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TheeXtensible Access Control Markup Language(XACML) is anXML-based standardmarkup languagefor specifyingaccess controlpolicies. The standard, published byOASIS, defines a declarative fine-grained, attribute-basedaccess controlpolicy language, an architecture, and a processing model describing how to evaluate access requests according to the rules defined in policies.[2]
XACML is primarily anattribute-based access controlsystem. In XACML, attributes – information about the subject accessing a resource, the resource to be addressed, and the environment – act as inputs for the decision of whether access is granted or not.[3]XACML can also be used to implementrole-based access control.[4]
In XACML, access control decisions to be taken are expressed as Rules. Each Rule comprises a series of conditions which decide whether a given request is approved or not. If a Rule is applicable to a request but the conditions within the Rule fail to evaluate, the result is Indeterminate. Rules are grouped together in Policies, and a PolicySet contains Policies and possibly other PolicySets. Each of these also includes a Target, a simple condition that determines whether it should be evaluated for a given request. Combining algorithms can be used to combine Rules and Policies with potentially differing results in various ways. XACML also supports obligations and advice expressions. Obligations specify actions which must be executed during the processing of a request, for example for logging. Advice expressions are similar, but may be ignored.[3]
XACML separates access control functionality into several components. Each operating environment in which access control is used has a Policy Enforcement Point (PEP) which implements the functionality to demand authorization and to grant or deny access to resources. These refer to an environment-independent and central Policy Decision Point (PDP) which actually makes the decision on whether access is granted. The PDP refers to policies stored in the Policy Retrieval Point (PRP). Policies are managed through a Policy Administration Point (PAP).[3]
Version 3.0 was ratified by OASIS in January 2013.[5]
Version 1.0 was ratified byOASISstandards organizationin 2003.[citation needed]
Version 2.0 was ratified by OASIS standards organization on February 1, 2005.[citation needed]
Version 3.0 was ratified by OASIS in January 2013.
Non-normative terminology (following RFC 2904, except for PAP)
(i.e. access to the resource is approved or rejected), and acts on the received decision
XACML is structured into 3 levels of elements:
A policy set can contain any number of policy elements and policy set elements. A policy can contain any number of rule elements.
Policies, policy sets, rules and requests all use subjects, resources, environments, and actions.
XACML provides a target, which is basically a set of simplified conditions for the subject, resource, and action that must be met for a policy set, policy, or rule to apply to a given request. Once a policy or policy set is found to apply to a given request, its rules are evaluated to determine the access decision and response.
In addition to being a way to check applicability, target information also provides a way to index policies, which is useful if you need to store many policies and then quickly sift through them to find which ones apply. When a request to access that service arrives, the PDP will know where to look for policies that might apply to this request because the policies are indexed based on their target constraints. Note that a target may also specify that it applies to any request.
Policy set, policy and rule can all contain target elements.
Conditions only exist in rules. Conditions are essentially an advanced form of a target which can use a broader range of functions and more importantly can be used to compare two or more attributes together, e.g. subject-id==doctor-id. With conditions, it is possible to implement segregation of duty checks or relationship-based access control.
Within XACML, a concept called obligations can be used. An obligation is a directive from the policy decision point (PDP) to the policy enforcement point (PEP) on what must be carried out before or after an access is approved. If the PEP is unable to comply with the directive, the approved accessmayormustnot be realized. The augmentation of obligations eliminates a gap between formal requirements and policy enforcement. An example of an obligation could look like this:
The XACML's obligation can be an effective way to meet formal requirements (non-repudiation for example) that can be hard to implement as access control rules. Furthermore, any formal requirements will be part of the access control policy as obligations and not as separate functions, which makes policies consistent and centralization of the IT environment easier to achieve.
Obligations can be used for "break-the-glass" scenarios or trust elevation ("you cannot transfer $1,000 without two-factor authentication - here is the link to the 2FA page").
In addition to obligations, XACML supports advice which are identical to obligations with the difference that a PEP is not obligated to enforce the advice (hence its name).
What happens in XACML if there are two rules (or policies) that contradict each other? Imagine for instance a first rule that would saymanagers can view documentsand a second rule that would sayno one can work before 9am. What if the request is about Alice trying to view a document at 8am? Which rule wins? This is what combining algorithms tell us. They help resolve conflicts.
XACML defines a number of combining algorithms that can be identified by aRuleCombiningAlgIdorPolicyCombiningAlgIdattribute of the <Policy> or <PolicySet> elements, respectively. The rule-combining algorithm defines a procedure for arriving at an access decision given the individual results of evaluation of a set of rules. Similarly, the policy-combining algorithm defines a procedure for arriving at an access decision given the individual results of evaluation of a set of policies.
XACML defines a long list of functions (close to 300) to manipulate and compare attributes to other attributes and values:
The functions and their identifiers are fully described in the standard. Functions are type-specific i.e. there is a function for string equality and a different one for integer equality.
Refer to the standard for a formal definition of these function.
Refer to the standard for a formal definition of these function.
The list of higher order functions is as listed below. For a formal definition, refer to the XACML standard.
http://docs.oasis-open.org/xacml/3.0/xacml-core-v3-schema-wd-17.xsd
XACML 3.0 introduces administrative delegation, the JSON Profile of XACML (request/response), the REST Profile of XACML, the Multiple Decision Profile of XACML, and many more.
The implementation of delegation is new in XACML 3.0. The delegation mechanism is used to support decentralized administration of access policies. It allows an authority (delegator) to delegate all or parts of its own authority or someone else's authority to another user (delegate) without any need to involve modification of the root policy.
This is because, in this delegation model, the delegation rights are separated from the access rights. These are instead referred to as administrative control policies. Access control and administrative policies work together as in the following scenario:
A partnership of companies' many services are protected by an access control system. The system implements the following central rules to protect its resources and to allow delegation:
(Attributes can be fetched from an external source, e.g. a LDAP catalog.)
When a consultant enters the corporation, a delegation can be issued locally by the consultant's supervisor, authorizing the consultant access to systems directly.
The delegator (the supervisor in this scenario) may only have the right to delegate a limited set of access rights to consultants.
Other new features of XACML 3.0 are listed athttp://www.webfarmr.eu/2010/07/enhancements-and-new-features-in-xacml-3-axiomatics/
The XACML TC is also publishing a list of changes here:http://wiki.oasis-open.org/xacml/DifferencesBetweenXACML2.0AndXACML3.0
This rule implements the use-it-lose-it access control paradigm. If a user does not log in for 30 days, then they lose access.
In pseudo-code: deny if currentDateTime > lastLogin + 30 days
This rule grants access if the current time is greater than 9am and less than 5pm.
The following contains an Obligation block. Obligations are statements that can be returned along with a decision to enrich the decision flow. In this example, the PEP must log that access was granted.
By default a PDP processes a single request at a time e.g. "Can Alice view item #1?". The PDP then replies with a single decision. At times, though, it is necessary to send multiple requests in one go e.g. "Can Alice view / edit / delete items #1, #2, #3?". The Multiple Decision Profile of XACML allows for this use case. The PDP will typically do the product of all combinations i.e. in the example aforementioned there will be 1 x 3 x 3 = 9 decisions returned in a single response.
The way to enable the MDP is to send an array of objects for any of the categories rather than an array of one object (or simply an object). For instance, AccessSubject is an object but Resource is an array of objects. The latter will trigger the MDP process in PDPs that support the profile. Note as well the use of the IncludeInResult attribute which tells the PDP to return the XACML attribute and its value in the response so that decisions can be correlated to the relevant attribute values.
In 2013 and 2014, the XACML Technical Committee focused on designing new profiles to facilitate developer integration. These include:
All three profiles were showcased at the Cloud Identity Summit 2014 in Monterey, California. Using these profiles, integrating fine-grained authorization into applications becomes much easier.
ALFA stands for Abbreviated Language for Authorization. It is a lightweight syntax used to implement policy-based access control policies. For examples refer to themain article.
The JSON profile of XACML simplifies the integration between the PEP and the PDP.
XACML is almost entirely apolicy definition languagebased onXMLandXSLT, defined by an openOASISspecification. The XACML specification does not cover the design or implementation of Policy Decision Point (PDP), only the policy language they consume. Manyproprietaryandopen-sourcePDPs use XACML as their policy definition language.
Open Policy Agent (OPA) is an open-source Policy Decision Point (PDP) implementation, capable of interpreting policy language to render policy decisions. OPA is a general-purpose PDP implementation which can be used for any scenario where a policy decision is required, much like PDP implementations that support the XACML specification.
OPA's policy definition language is (Rego), which is a JSON-based, Turing-incomplete language based on Datalog.
Policies written in XACML can be translated to Rego, and vice-versa.
SAMLis an identity SSO and federation standard used for authentication. SAML is used as a common identity token format between different applications. SAML and XACML are both defined byOASIS. SAML and XACML were designed to interoperate where SAML is used to carry identity information / virtual identities and XACML is used to drive the access control logic through policies.
OAuth 2.0is considered to be an authorization standard. It differs from XACML though in its origin, its purpose, and its applications. OAuth is about:
XACML does not handle user approval or delegated access or password management. XACML simply provides:
XACML and OAuth can be combined to deliver a more comprehensive approach to authorization.
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https://en.wikipedia.org/wiki/XACML
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OpenAthensis anidentity and access managementservice, supplied byJisc, a Britishnot-for-profitinformation technology services company.Identity provider(IdP) organisations can keepusernamesin the cloud, locally or both. Integration withADFS,LDAPorSAMLis supported.[1]
OpenAthens for Publishers[2]software forservice providerssupports multiple platforms andfederations.
Technically, the service providesdeep packet inspectionproxying (in a similar manner toEZproxy) andSAML-based federation,[3]as well as various on-boarding services for institutions, consortia and vendors.
With its origins in aUniversity of Bathinitiative to reduce IT procurement costs for itself and other universities, the Athens project was conceived in 1996. Spun off from Bath University through the vehicle of charitable status, Eduserv was established as a not-for-profit organisation in 1999.[4]
The service was originally namedAthenaafter the Greek goddess of knowledge and learning; it is rumoured that the name change was partially caused by a common typo, but it was actually due to the name Athena being already trademarked (EU000204735).[5]It launched as 'Athens' in 1997 (UK00002153200).[6]After JISC decided to supportShibbolethrather than Athens in 2008, Eduserv launched a federated version of Athens as 'OpenAthens'[7](EU013713821).[8]
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https://en.wikipedia.org/wiki/OpenAthens
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MozillaPersonawas a decentralizedauthenticationsystem for the web, based on the open BrowserID protocol[1]prototyped byMozilla[2]and standardized byIETF.[3]It was launched in July 2011, but after failing to achieve traction, Mozilla announced in January 2016 plans to decommission the service by the end of the year.[4]
Persona was launched in July 2011[5]and shared some of its goals with some similar authentication systems likeOpenIDorFacebook Connect, but it was different in several ways:
The privacy goal was motivated by the fact that the identity provider does not know which website the user is identifying on.[6]It was first released in July 2011 and fully deployed byMozillaon its own websites in January 2012.[7]
In March 2014, Mozilla indicated it was dropping full-time developers from Persona and moving the project to community ownership. Mozilla indicated, however, that it had no plans to decommission Persona and would maintain some level of involvement such as in maintenance and reviewingpull requests.[8]
Persona services are shut down since November 30, 2016.[9]
Persona was inspired by theVerifiedEmailProtocol[10][11]which is now known as theBrowserIDprotocol.[12]It uses any useremail addressto identify its owner. This protocol involves the browser, an identity provider, and any compliant website.
The browser stores a list of user verified email addresses (certificates issued by the identity providers), and demonstrates the user's ownership of the addresses to the website usingcryptographicproof.[13]
The certificates must be renewed every 24 hours by logging into the identity provider (which will usually mean entering the email and a password in a Web form on the identity provider's site). Once done, they will be usable for authenticating to websites with the same browser for the rest of the day, without entering passwords again (single sign-on).[14]
The decentralization aspects of the protocol reside in the theoretical support of any identity provider service, while in practice it seems to rely mainly on Mozilla's servers currently (which may in turn delegate email address verification, seeidentity bridgingbelow). However, even if the protocol heavily relies on a central identity provider, this central actor only knows when browsers renew certificates, and cannot in principle monitor where the certificates will be used.
Mozilla announced "identity bridging" support for Persona in July 2013. As they describe on their blog:
"Traditionally ... Mozilla would send you an email and ask you to click on the confirmation link it contained. With Identity Bridging, Persona learned a new trick; instead of sending confirmation emails, Persona can ask you to verify your identity via your email provider’s existingOpenIDorOAuthgateway."[15]
This announcement included support for existing users of the Yahoo Mail service. In August 2013, Mozilla announced support for Identity Bridging with all Gmail accounts. They wrote in this additional announcement that "combined with our Identity Bridge for Yahoo, Persona now natively supports more than 700,000,000 active email users. That covers roughly 60–80% of people on most North American websites."[16]
Persona relies heavily on the JavaScript client-side program running in the user's browser, making it widely usable.
Support of authentication to Web applications via Persona can be implemented byCMSssuch asDrupal,[17]Serendipity,[18]WordPress,[19]Tiki,[20]orSPIP. There is also support for Persona in thePhonegap[21]platform (used for compilingHTML5apps into mobile apps).Mozillaprovides its own Persona server at persona.org.[22]It is also possible to set up your own Persona identity provider,[23]providingfederated identity.
Notable sites implementing Persona includeTing,[24]The TimesCrossword, andVoost.[25]
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https://en.wikipedia.org/wiki/Mozilla_Persona
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TheCentral Authentication Service(CAS) is asingle sign-onprotocol for theweb.[1]Its purpose is to permit a user to access multiple applications while providing their credentials (such as user ID and password) only once. It also allows web applications to authenticate users without gaining access to a user's security credentials, such as a password. The nameCASalso refers to asoftware packagethat implements this protocol.
The CAS protocol involves at least three parties: aclientweb browser, the webapplicationrequesting authentication, and theCAS server. It may also involve aback-end service, such as a database server, that does not have its own HTTP interface but communicates with a web application.
When the client visits an application requiring authentication, the application redirects it to CAS. CAS validates the client's authenticity, usually by checking a username and password against a database (such asKerberos,LDAPorActive Directory).
If the authentication succeeds, CAS returns the client to the application, passing along aservice ticket. The application then validates the ticket by contacting CAS over a secure connection and providing its own service identifier and the ticket. CAS then gives the application trusted information about whether a particular user has successfully authenticated.
CAS allows multi-tier authentication viaproxy address. A cooperatingback-endservice, like a database or mail server, can participate in CAS, validating the authenticity of users via information it receives from web applications. Thus, a webmail client and a webmail server can all implement CAS.
CAS was conceived and developed byShawn BayernofYale UniversityTechnology and Planning. It was later maintained by Drew Mazurek at Yale. CAS 1.0 implemented single-sign-on. CAS 2.0 introduced multi-tier proxy authentication. Several other CAS distributions have been developed with new features.
In December 2004, CAS became a project of theJava in Administration Special Interest Group (JASIG), which is as of 2008 responsible for its maintenance and development. Formerly called "Yale CAS", CAS is now also known as "Jasig CAS". In 2010, Jasig entered into talks with the Sakai Foundation to merge the two organizations. The two organizations were consolidated as Apereo Foundation in December 2012.
In December 2006, theAndrew W. Mellon Foundationawarded Yale its First Annual Mellon Award for Technology Collaboration, in the amount of $50,000, for Yale's development of CAS.[2]At the time of that award CAS was in use at "hundreds of university campuses (among other beneficiaries)".
In April 2013, CAS Protocol specification 3.0 was released.[3]
The Apereo CAS server that is the reference implementation of the CAS protocol today supports the following features:
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https://en.wikipedia.org/wiki/Central_Authentication_Service
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IndieAuthis an open standarddecentralizedauthenticationprotocolthat usesOAuth2.0 and enables services to verify the identity of a user represented by aURL, as well as to obtain anaccess token, that can be used to access resources under the control of the user.[1][2][3]
IndieAuth is developed in theIndieWebcommunity and was published as aW3CNote.[3]It was published as a W3C Note by theSocial Web Working Groupdue to lacking the time needed to formally progress it to aW3C recommendation, despite having several interoperable implementations.[4]
ThisWorld Wide Web–related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/IndieAuth
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Aninformation card(ori-card) is a personal digital identity that people can use online, and the key component of an identity metasystem. Visually, each i-card has a card-shaped picture and a card name associated with it that enable people to organize their digital identities and to easily select one they want to use for any given interaction. The information card metaphor has been implemented by identity selectors likeWindows CardSpace, DigitalMe orHiggins Identity Selector.
An identity metasystem is an interoperable architecture fordigital identitythat enables people to have and employ a collection of digital identities based on multiple underlying technologies, implementations, and providers. Using this approach, customers can continue to use their existing identity infrastructure investments, choose the identity technology that works best for them, and more easily migrate from old technologies to new technologies without sacrificing interoperability with others. The identity metasystem is based upon the principles in "The Laws of Identity".[1]
There are three participants indigital identityinteractions using information cards:[citation needed]
An identity selector is used to store, manage, and use their digital identities. Examples of identity selectors are Microsoft'sWindows CardSpace, theBandit Project's DigitalMe,[2]and several kinds of Identity Selectors from theEclipse Foundation's Higgins project.
An identity selector performs the following user-centric identity management tasks:
An identity selector may also allow the user to manage (e.g. create, review, update, and delete cards within) their portfolio of i-cards.
There are five key components to an identity metasystem:
Using i-cards, users can authenticate without needing a username and password for every website; instead, at sites accepting them, they can log in with an i-card, which may be used at multiple sites.
Each information card utilizes a distinct pair-wise digital key for every realm where a key is requested. A realm may be a single site or a set of related sites all sharing the same target scope information when requesting an information card. The use of distinct pair-wise keys per realm means that even if a person is tricked into logging into an imposter site with an i-card, a different key would be used at that site than the site that the imposter was trying to impersonate; no shared secret is released.
Furthermore, many identity selectors provide a means ofphishingdetection, where theHTTPScertificate of the relying party site is checked and compared against a list of the sites at which the user has previously used an information card. When a new site is visited, the user is informed that they have not previously used a card there.
The Identity Selector Interoperability Profile v 1.5[3](or OASIS IMI v1.0 Committee Draft)[4]specifies two types of information cards that an identity selector must support.
TheHiggins projectis defining two new kinds of i-cards as well:
However the Information Card format allows for custom types; The Bandit project demonstrated prototype managed cards backed byOpenIDsat theNovell BrainShareconference in March 2007.
The first kind of personal Information cards were also introduced as part of Microsoft’s Windows CardSpace software in November 2006. Their behavior is also defined by the same documents covering the Microsoft-defined managed cards (see above).
Summary of characteristics:
The first kind of managed card was introduced as part of Microsoft’s Windows CardSpace software in November 2006. The behavior, file format and interoperability characteristics of these kinds of managed cards are defined by Microsoft documents such as the Identity Selector Interoperability Profile v 1.5[3](or OASIS IMI v1.0 Committee Draft;[4]see self-issued.info[5]for a more complete list), in combination with open standards including WS-Trust[6]and others.
Summary of characteristics:
I-cards issued by third parties can employ any of four methods for the user to authenticate himself as the card owner:
Additional methods could also be implemented by future identity selectors and identity providers.
Managed i-cards can be auditing, non-auditing, or auditing-optional:
Relationship cards are under development by the Higgins project (see the report by Paul Trevithick).[7]
Summary of characteristics:
Conceptually a managed card is essentially a human-friendly "pointer" to a Token Service—a web service (e.g. aSTS) from which security tokens can be requested. A security token is a set of attribute assertions (aka claims) about some party that is cryptographically signed by the issuer (the token service acting as the authority). An r-card, contains a second "pointer" that points to a data entity whose attribute's values (i) shared by all parties to the r-card and (ii) form the underlying attributes that are consumed by the r-card issuer's STS and provide the values of the claims that this STS makes. By including this second "pointer" on the r-card, r-card holders have the potential to access and update some subset of these underlying attributes. The card issuer maintains an access control policy to control who has what level of access.
This second pointer is an Entity UDI[8]—a reference to anEntityobject in the Higgins Context Data Model.[9]Entity UDIs may be dereferenced and the underlying Entity's attributes accessed by using the Higgins project'sIdentity Attribute Service.[10]Once resolved, consumers of this service can inspect, and potentially modify the attributes of the entity as well as get its schema as described inWeb Ontology Language(OWL).
In addition to basic identity attribute values like strings and numbers, the data entity referred to by an r-card can have complex attribute values consisting of aggregates of basic attribute types as well as UDI links to other entities.
Beyond being used to log into sites, Information Cards can also facilitate other kinds of interactions. The Information Card model provides great flexibility because cards can be used to convey any information from an Identity Provider to a Relying Party that makes sense to both of them and that the person is willing to release. The data elements carried in i-cards are called Claims.
One possible use of claims is online age verification, with Identity Providers providing proof-of-age cards, and RPs accepting them for purposes such as online wine sales; other attributes could be verified as well. Another isonline payment, where merchants could accept online payment cards from payment issuers, containing only the minimal information needed to facilitate payment. Role statements carried by claims can be used for access control decisions by Relying Parties.
The protocols needed to build Identity Metasystem components can be used by anyone for any purpose with no licensing cost and interoperable implementations can be built using only publicly available documentation. Patent promises have been issued by Microsoft,[11]IBM,[12]and others ensuring that the protocols underlying the Identity Metasystem can be freely used by all.
The Information Cards defined by the Identity Selector Interoperability Profile v 1.5[3](or OASIS IMI v1.0 Committee Draft)[4]are based on open, interoperable communication standards. Interoperable i-card components have been built by dozens of companies and projects for platforms including Windows, Mac OS, and Linux, plus a prototype implementation for phones. Together, these components implement an interoperable Identity Metasystem. Information Cards can be used to provide identities both for Web sites and Web Services applications.
Several interoperability testing events for i-cards have been sponsored by OSIS[13]and the Burton Group,[14]one was at the Interop at the October 2007 European Catalyst Conference in Barcelona[15]and the most recent was at RSA 2008. These events are helping to ensure that the different Information Card software components being built by the numerous participants in the Identity Metasystem work well together.
The protocols needed to build Information Card implementations based on the Identity Selector Interoperability Profile v 1.5[3](or OASIS IMI v1.0 Committee Draft)[4]can be used by anyone for any purpose at no cost and interoperable implementations can be built using only publicly available documentation. Patent promises have been issued by Microsoft,[11]IBM,[12]and others, ensuring that this Information Card technology is freely available to all.
In June 2008, industry leaders including Equifax, Google, Microsoft, Novell, Oracle, PayPal and others created theInformation Card Foundationin order to advance the use of the Information Card metaphor as a key component of an open, interoperable, royalty-free, user-centric identity layer spanning both the enterprise and the Internet.
In his report on the Interop at the June 2007 Catalyst Conference in San Francisco,[16]analyst Bob Blakley wrote:
The interop event was a milestone in the maturation of user-centric identity technology. Prior to the event, there were some specifications, one commercial product, and a number of open-source projects. After the event, it can accurately be said that there is a running Identity Metasystem.
The term "information card" was introduced by Microsoft in May 2005 as a name for the visual information card metaphor to be introduced in its forthcoming Windows CardSpace software. Until early 2006, information cards were also sometimes referred to by the code-name “InfoCard”, which was not a name that was freely available for all to use. The name information card was specifically chosen as one that would be freely available for all to use, independent of any product or implementation. The name “information card” is not trademarked and is so generic as to not be trademarkable.
The term i-card was introduced at the June 21, 2006, Berkman/MIT Identity Mashup conference.[17][18]The intent was to define a term that was not associated with any industry TM or other IP or artifact. At the time, Microsoft had not yet finished applying the Open Specification Promise[11]to the protocols underlying Windows CardSpace and there was also a misunderstanding that the term information card was not freely available for use by all, so to be conservative, the term i-card was introduced.
Mike Jones, of Microsoft, explained to participants of a session at IIW 2007b[19]that Microsoft always intended the term information card to be used generically to describe all kinds of information cards and to be freely usable by all, and tried to correct the earlier misunderstanding that the term might apply only to the kinds of information cards originally defined by Microsoft. He made the case that the industry would be better served by having everyone use the common term information card, than having two terms in use with the same meaning, since there remains no legal or technical reason for different terms. In this case the term i-card would become just the short form of information card, just likee-mailhas become the short form of electronic mail.
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https://en.wikipedia.org/wiki/Information_card
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TheLiberty Alliance Projectwas an organization formed in September 2001 to establish standards, guidelines and best practices foridentity managementin computer systems.
It grew to more than 150 organizations, including technology vendors, consumer-facing companies, educational organizations and governments.
It released frameworks for federation, identity assurance, anIdentity Governance Framework, and Identity Web Services.
By 2009, theKantara Initiativetook over the work of the Liberty Alliance.
The group was originally conceived and named byJeff Veis, atSun Microsystemsbased inMenlo Park, California.[1]The initiative's goal, which was personally promoted byScott McNealyof Sun, was to unify technology, commercial and government organizations to create a standard for federated, identity-based Internet applications as an alternative to technology appearing in the marketplace controlled by a single entity such asMicrosoft'sPassport.[2]Another Microsoft initiative,HailStorm, was renamed My Services but quietly shelved by April 2002.[3]Sun positioned the group as independent, andEric C. DeanofUnited Airlinesbecame its president.[4]
In July 2002, the alliance announced Liberty Identity Federation (ID-FF) 1.0.[5]At that time, several member companies announced upcoming availability of Liberty-enabled products.
Liberty Federation allowed consumers and users of Internet-based services and e-commerce applications to authenticate and sign-on to a network or domain once from any device and then visit or take part in services from multiple Websites. This federated approach did not require the user to re-authenticate and can support privacy controls established by the user.
The Liberty Alliance subsequently released two more versions of the Identity Federation Framework, and then in November 2003, Liberty contributed its final version of the specification, ID-FF 1.2, toOASIS.[6]This contribution formed the basis forSAML 2.0. By 2007, industry analyst firmGartnerclaimed that SAML had gained wide acceptance in the community.[7]
Liberty Alliance, releasing the Liberty Identity Web Services Framework (ID-WSF) in April 2004 for deploying and managing identity-based web services. Applications includedgeolocation, contact book, calendar, mobile messaging and People Service, for managing social applications such as bookmarks, blogs, calendars, photo sharing and instant messaging in a secure and privacy-respecting federated social network.
In a 2008 marketing report recommended considering it for federation.[8]
The alliance introduced a certification program in 2003, designed to test commercial and open source products against published standards to assure base levels of interoperability between products. In 2007, the USGeneral Services Administrationbegan requiring this certification for participating in the US E-Authentication Identity Federation.[9]
In January 2007, the alliance announced a project foropen-source softwaredevelopers building identity-based applications. OpenLiberty.org was a portal where developers can collaborate and access tools and information to develop applications based on alliance standards.[10]In November 2008, OpenLiberty released an open sourceapplication programming interfacecalled ArisID.[11]
In February 2007Oracle Corporationcontributed theIdentity Governance Frameworkto the alliance,[12]which released the first version publicly in July 2007.[13]The Identity Governance Framework defined how identity related information is used, stored, and propagated using protocols such asLDAP, Security Assertion Markup Language,WS-Trust, and ID-WSF.
The Liberty Alliance began work on itsidentity assuranceframework in 2008. The Identity Assurance Framework (IAF) detailed four identity assurance levels designed to link trusted identity-enabled enterprise, social networking and Web applications together based on business rules and security risks associated with each level. The four levels of assurance were outlined by a 2006 document from the USNational Institute of Standards and Technology.[14]The level of assurance provided is measured by the strength and rigor of the identity proofing process, the credential's strength, and the management processes the service provider applies to it.
These four assurance levels were adopted by UK, Canada, and USA government services.
In 2007 the Liberty Alliance helped to found theProject Concordia, an independent initiative for harmonization identity specifications. It was active through 2008.[15]
The alliance wrote papers on business and policy aspects of identity management.[16]It hosted meetings in 2007 and 2008 to promote itself.[17]
Management board members includedAOL,British Telecom,Computer Associates(CA),Fidelity Investments,Intel,Internet Society(ISOC),Novell,Nippon Telegraph and Telephone(NTT), Vodafone, Oracle Corporation and Sun Microsystems.
As described above,Liberty contributed Identity Federation Framework (ID-FF) 1.2 to OASISin November 2003. For the record, here is a complete list of contributed ID-FF 1.2 documents:
Only the archived PDF files are individually addressable on the Liberty Alliance web site. (The original contributed documents are lost.) To obtain copies of the remaining archived files, download both theLiberty ID-FF 1.2 archiveand theLiberty 1.1 support archive.
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https://en.wikipedia.org/wiki/Liberty_Alliance
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Light-weight Identity(LID), orLight Identity Management(LIdM), is anidentity managementsystem foronline digital identitiesdeveloped in part byNetMesh. It was first published in early 2005, and is the original URL-based identity system, later followed byOpenID. LID usesURLsas a verification of the user's identity, and makes use of several open-source protocols such as OpenID,Yadis, andPGP/GPG.
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https://en.wikipedia.org/wiki/Light-weight_Identity
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Security Assertion Markup Language(SAML, pronouncedSAM-el,/ˈsæməl/)[1]is anopen standardfor exchangingauthenticationandauthorizationdata between parties, in particular, between anidentity providerand aservice provider. SAML is anXML-basedmarkup languagefor security assertions (statements that service providers use to make access-control decisions). SAML is also:
An important use case that SAML addresses isweb-browsersingle sign-on(SSO). Single sign-on is relatively easy to accomplish within asecurity domain(usingcookies, for example) but extending SSO across security domains is more difficult and resulted in the proliferation of non-interoperable proprietary technologies. The SAML Web Browser SSO profile was specified and standardized to promote interoperability.[2]In practice, SAML SSO is most commonly used for authentication into cloud-based business software.[3]
The SAML specification defines three roles: the principal (typically a human user), theidentity provider(IdP) and theservice provider(SP). In the primary use case addressed by SAML, the principal requests a service from the service provider. The service provider requests and obtains an authentication assertion from the identity provider. On the basis of this assertion, the service provider can make anaccess controldecision, that is, it can decide whether to perform the service for the connected principal.
At the heart of the SAML assertion is a subject (a principal within the context of a particular security domain) about which something is being asserted. The subject is usually (but not necessarily) a human. As in the SAML 2.0 Technical Overview,[4]the terms subject and principal are used interchangeably in this document.
Before delivering the subject-based assertion from Identity Provider to the Service Provider, the Identity Provider may request some information from the principal (such as a user name and password) in order to authenticate the principal. SAML specifies the content of the assertion that is passed from the Identity Provider to the Service Provider. In SAML, one Identity Provider may provide SAML assertions to many Service Providers. Similarly, one Service Provider (SP) may rely on and trust assertions from many independent Identity Providers (IdP).[5]
SAML does not specify the method of authentication at the identity provider. The IdP may use a username and password, or some other form of authentication, includingmulti-factor authentication. A directory service such asRADIUS,LDAP, orActive Directorythat allows users to log in with a user name and password is a typical source of authentication tokens at an identity provider.[6]The popular Internet social networking services also provide identity services that in theory could be used to support SAML exchanges.
TheOrganization for the Advancement of Structured Information Standards (OASIS)Security Services Technical Committee (SSTC), which met for the first time in January 2001, was chartered "to define an XML framework for exchanging authentication and authorization information."[7]To this end, the following intellectual property was contributed to the SSTC during the first two months of that year:
Building on these initial contributions, in November 2002 OASIS announced the Security Assertion Markup Language (SAML) 1.0 specification as an OASIS Standard.[8]
Meanwhile, theLiberty Alliance, a large consortium of companies, non-profit and government organizations, proposed an extension to the SAML standard called the Liberty Identity Federation Framework (ID-FF).[9]Like its SAML predecessor, Liberty ID-FF proposed a standardized, cross-domain, web-based, single sign-on framework. In addition, Liberty described acircle of trustwhere each participating domain is trusted to accurately document the processes used to identify a user, the type of authentication system used, and any policies associated with the resulting authentication credentials. Other members of the circle of trust could then examine these policies to determine whether to trust such information.[10]
While Liberty was developing ID-FF, the SSTC began work on a minor upgrade to the SAML standard. The resulting SAML 1.1 specification was ratified by the SSTC in September 2003. Then, in November of that same year,Liberty contributed ID-FF 1.2 to OASIS, thereby sowing the seeds for the next major version of SAML. In March 2005, SAML 2.0 was announced as an OASIS Standard. SAML 2.0 represents the convergence of Liberty ID-FF and proprietary extensions contributed by theShibbolethproject, as well as early versions of SAML itself. Most SAML implementations support v2.0 while many still support v1.1 for backward compatibility. By January 2008, deployments of SAML 2.0 became common in government, higher education, and commercial enterprises worldwide.[10]
SAML has undergone one minor and one major revision since 1.0.
The Liberty Alliance contributed its Identity Federation Framework (ID-FF) to the OASIS SSTC in September 2003:
Versions 1.0 and 1.1 of SAML are similar even though small differences exist.,[11]however, the differences between SAML 2.0 and SAML 1.1 are substantial. Although the two standards address the same use case, SAML 2.0 is incompatible with its predecessor.
Although ID-FF 1.2 was contributed to OASIS as the basis of SAML 2.0, there are some important differences between SAML 2.0 and ID-FF 1.2. In particular, the two specifications, despite their common roots, are incompatible.[10]
SAML is built upon a number of existing standards:
SAML defines XML-based assertions and protocols, bindings, and profiles. The termSAML Corerefers to the general syntax and semantics of SAML assertions as well as the protocol used to request and transmit those assertions from one system entity to another.SAML protocolrefers towhatis transmitted, nothow(the latter is determined by the choice of binding). So SAML Core defines "bare" SAML assertions along with SAML request and response elements.
ASAML bindingdetermines how SAML requests and responses map onto standard messaging or communications protocols. An important (synchronous) binding is the SAML SOAP binding.
ASAML profileis a concrete manifestation of a defined use case using a particular combination of assertions, protocols and bindings.
A SAMLassertioncontains a packet of security information:
Loosely speaking, a relying party interprets an assertion as follows:
AssertionAwas issued at timetby issuerRregarding subjectSprovided conditionsCare valid.
SAML assertions are usually transferred from identity providers to service providers. Assertions containstatementsthat service providers use to make access-control decisions. Three types of statements are provided by SAML:
Authentication statementsassert to the service provider that the principal did indeed authenticate with the identity provider at a particular time using a particular method of authentication. Other information about the authenticated principal (called theauthentication context) may be disclosed in an authentication statement.
Anattribute statementasserts that a principal is associated with certain attributes. Anattributeis simply aname–value pair. Relying parties use attributes to make access-control decisions.
Anauthorization decision statementasserts that a principal is permitted to perform actionAon resourceRgiven evidenceE. The expressiveness of authorization decision statements in SAML is intentionally limited. More-advanced use cases are encouraged to useXACMLinstead.
A SAMLprotocoldescribes how certain SAML elements (including assertions) are packaged within SAML request and response elements, and gives the processing rules that SAML entities must follow when producing or consuming these elements. For the most part, a SAML protocol is a simple request-response protocol.
The most important type of SAML protocol request is called aquery. A service provider makes a query directly to an identity provider over a secure back channel. Thus query messages are typically bound to SOAP.
Corresponding to the three types of statements, there are three types of SAML queries:
The result of an attribute query is a SAML response containing an assertion, which itself contains an attribute statement. See the SAML 2.0 topic foran example of attribute query/response.
Beyond queries, SAML 1.1 specifies no other protocols.
SAML 2.0 expands the notion ofprotocolconsiderably. The following protocols are described in detail in SAML 2.0 Core:
Most of these protocols are new inSAML 2.0.
A SAMLbindingis a mapping of a SAML protocol message onto standard messaging formats and/or communications protocols. For example, the SAML SOAP binding specifies how a SAML message is encapsulated in a SOAP envelope, which itself is bound to an HTTP message.
SAML 1.1 specifies just one binding, the SAML SOAP Binding. In addition to SOAP, implicit in SAML 1.1 Web Browser SSO are the precursors of the HTTP POST Binding, the HTTP Redirect Binding, and the HTTP Artifact Binding. These are not defined explicitly, however, and are only used in conjunction with SAML 1.1 Web Browser SSO. The notion of binding is not fully developed until SAML 2.0.
SAML 2.0 completely separates the binding concept from the underlying profile. In fact, there is a brandnew binding specification in SAML 2.0that defines the following standalone bindings:
This reorganization provides tremendous flexibility: taking just Web Browser SSO alone as an example, a service provider can choose from four bindings (HTTP Redirect, HTTP POST and two flavors of HTTP Artifact), while the identity provider has three binding options (HTTP POST plus two forms of HTTP Artifact), for a total of twelve possible deployments of the SAML 2.0 Web Browser SSO Profile.
A SAMLprofiledescribes in detail how SAML assertions, protocols, and bindings combine to support a defined use case. The most important SAML profile is the Web Browser SSO Profile.
SAML 1.1 specifies two forms of Web Browser SSO, the Browser/Artifact Profile and the Browser/POST Profile. The latter passes assertionsby valuewhereas Browser/Artifact passes assertionsby reference. As a consequence, Browser/Artifact requires a back-channel SAML exchange over SOAP. In SAML 1.1, all flows begin with a request at the identity provider for simplicity. Proprietary extensions to the basic IdP-initiated flow have been proposed (byShibboleth, for example).
The Web Browser SSO Profile was completely refactored for SAML 2.0. Conceptually, SAML 1.1 Browser/Artifact and Browser/POST are special cases of SAML 2.0 Web Browser SSO. The latter is considerably more flexible than its SAML 1.1 counterpart due to the new "plug-and-play" binding design of SAML 2.0. Unlike previous versions, SAML 2.0 browser flows begin with a request at the service provider. This provides greater flexibility, but SP-initiated flows naturally give rise to the so-calledIdentity Provider Discoveryproblem, the focus of much research today. In addition to Web Browser SSO, SAML 2.0 introduces numerous new profiles:
Aside from the SAML Web Browser SSO Profile, some important third-party profiles of SAML include:
The SAML specifications recommend, and in some cases mandate, a variety of security mechanisms:
Requirements are often phrased in terms of (mutual) authentication, integrity, and confidentiality, leaving the choice of security mechanism to implementers and deployers.
The primary SAML use case is calledWeb Browser Single Sign-On (SSO). A user utilizes auser agent(usually a web browser) to request a web resource protected by a SAMLservice provider. The service provider, wishing to know the identity of the requesting user, issues an authentication request to a SAMLidentity providerthrough the user agent. The resulting protocol flow is depicted in the following diagram.
In SAML 1.1, the flow begins with a request to the identity provider's inter-site transfer service at step 3.
In the example flow above, all depicted exchanges arefront-channel exchanges, that is, an HTTP user agent (browser) communicates with a SAML entity at each step. In particular, there are noback-channel exchangesor direct communications between the service provider and the identity provider. Front-channel exchanges lead to simple protocol flows where all messages are passedby valueusing a simple HTTP binding (GET or POST). Indeed, the flow outlined in the previous section is sometimes called theLightweight Web Browser SSO Profile.
Alternatively, for increased security or privacy, messages may be passedby reference. For example, an identity provider may supply a reference to a SAML assertion (called anartifact) instead of transmitting the assertion directly through the user agent. Subsequently, the service provider requests the actual assertion via a back channel. Such a back-channel exchange is specified as aSOAPmessage exchange (SAML over SOAP over HTTP). In general, any SAML exchange over a secure back channel is conducted as a SOAP message exchange.
On the back channel, SAML specifies the use of SOAP 1.1. The use of SOAP as a binding mechanism is optional, however. Any given SAML deployment will choose whatever bindings are appropriate.
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Shibbolethis asingle sign-onlog-in system for computer networks and theInternet. It allows people to sign in using just one identity to various systems run by federations of different organizations or institutions. The federations are often universities or public service organizations.
The ShibbolethInternet2middlewareinitiative created anarchitectureandopen-sourceimplementation foridentity managementandfederated identity-basedauthenticationandauthorization(oraccess control) infrastructure based onSecurity Assertion Markup Language(SAML). Federated identity allows the sharing of information about users from one security domain to the other organizations in a federation. This allows for cross-domain single sign-on and removes the need for content providers to maintain usernames and passwords.Identity providers(IdPs) supply user information, while service providers (SPs) consume this information and give access to secure content.
The Shibboleth project grew out of Internet2. Today, the project is managed by the Shibboleth Consortium. Two of the most popular software components managed by the Shibboleth Consortium are the Shibboleth Identity Provider and the Shibboleth Service Provider, both of which are implementations ofSAML.
The project was named after anidentifying passphraseused in theBible(Judges12:4–6) becauseEphraimiteswere not able to pronounce "sh".
The Shibboleth project was started in 2000 to facilitate the sharing of resources between organizations with incompatibleauthentication and authorization infrastructures.Architectural workwas performed for over a year prior to any software development. After development and testing, Shibboleth IdP 1.0 was released in July 2003.[1]This was followed by the release of Shibboleth IdP 1.3 in August 2005.
Version 2.0 of the Shibboleth software was a major upgrade released in March 2008.[2]It included both IdP and SP components, but, more importantly, Shibboleth 2.0 supported SAML 2.0.
The Shibboleth and SAML protocols were developed during the same timeframe. From the beginning, Shibboleth was based on SAML, but, where SAML was found lacking, Shibboleth improvised, and the Shibboleth developers implemented features that compensated for missing features inSAML 1.1. Some of these features were later incorporated intoSAML 2.0, and, in that sense, Shibboleth contributed to the evolution of the SAML protocol.
Perhaps the most important contributed feature was the legacy Shibboleth AuthnRequest protocol. Since the SAML 1.1 protocol was inherently an IdP-first protocol, Shibboleth invented a simple HTTP-based authentication request protocol that turned SAML 1.1 into an SP-first protocol. This protocol was first implemented in Shibboleth IdP 1.0 and later refined in Shibboleth IdP 1.3.
Building on that early work, theLiberty Allianceintroduced a fully expanded AuthnRequest protocol into the Liberty Identity Federation Framework. Eventually, Liberty ID-FF 1.2 was contributed to OASIS, which formed the basis for the OASIS SAML 2.0 Standard.[importance?]
Shibboleth is a web-based technology that implements the HTTP/POST artifact and attribute push profiles ofSAML, including both Identity Provider (IdP) and Service Provider (SP) components. Shibboleth 1.3 has its own technical overview,[3]architectural document,[4]and conformance document[5]that build on top of the SAML 1.1 specifications.
In the canonical use case:
Shibboleth supports a number of variations on this base case, including portal-style flows whereby the IdP mints an unsolicited assertion to be delivered in the initial access to the SP, and lazy session initiation, which allows an application to trigger content protection through a method of its choice as required.
Shibboleth 1.3 and earlier do not provide a built-inauthenticationmechanism, but any Web-based authentication mechanism can be used to supply user data for Shibboleth to use. Common systems for this purpose includeCASorPubcookie. The authentication and single-sign-on features of the Java container in which the IdP runs (Tomcat, for example) can also be used.
Shibboleth 2.0 builds onSAML 2.0standards. The IdP in Shibboleth 2.0 has to do additional processing in order to support passive and forced authentication requests in SAML 2.0. The SP can request a specific method of authentication from the IdP. Shibboleth 2.0 supports additional encryption capacity.
Shibboleth's access control is performed by matching attributes supplied by IdPs against rules defined by SPs. An attribute is any piece of information about a user, such as "member of this community", "Alice Smith", or "licensed under contract A". User identity is considered an attribute, and is only passed when explicitly required, which preserves user privacy. Attributes can be written in Java or pulled from directories and databases. StandardX.520attributes are most commonly used, but new attributes can be arbitrarily defined as long as they are understood and interpreted similarly by the IdP and SP in a transaction.
Trust between domains is implemented using public key cryptography (often simplyTLSserver certificates) and metadata that describes providers. The use of information passed is controlled through agreements. Federations are often used to simplify these relationships by aggregating large numbers of providers that agree to use common rules and contracts.
Shibboleth is open-source and provided under the Apache 2 license. Many extensions have been contributed by other groups.[citation needed]
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SQRL(pronounced "squirrel")[2]orSecure, Quick, Reliable Login(formerlySecureQRLogin) is adraftopen standardfor securewebsiteloginandauthentication. Thesoftwaretypically uses alink of the schemesqrl://or optionally aQR code, where a user identifies via apseudonymouszero-knowledge proofrather than providing auser IDandpassword. This method is thought to be impervious to abrute-forcepassword attack ordata breach. It shifts the burden of security away from the party requesting the authentication and closer to the operating-systemimplementationof what is possible on thehardware, as well as to the user. SQRL was proposed bySteve Gibsonof Gibson Research Corporation in October 2013 as a way to simplify the process ofauthenticationwithout the risk of revelation of information about the transaction to athird party.
TheacronymSQRL was coined by Steve Gibson and the protocol drafted, discussed and analyzed in-depth, by himself and a community ofInternet securityenthusiastson thenews.grc.comnewsgroupsand during his weeklypodcast,Security Now!, on October 2, 2013. Within two days of the airing of this podcast, theW3Cexpressed interest in working on the standard.[3]
Google Cloud Platformdevelopers Ian Maddox and Kyle Moschetto mentioned SQRL in their document "Modern Password Security for System Designers".[4]
A thesis on SQRL analyzed and found that "it appears to be an interesting approach, both in terms of the envisioned user experience as well as the underlying cryptography. SQRL is mostly combining well established cryptography in a novel way."[5]
The protocol is an answer to a problem ofidentityfragmentation. It improves on protocols such asOAuthandOpenIDby not requiring athird partyto broker the transaction, and by not giving a server any secrets to protect, such as username and password.
Additionally, it provides a standard that can be freely used to simplify the login processes available topassword managerapplications. More importantly, the standard is open so no one company can benefit from owning the technology. According to Gibson's website,[2]such a robust technology should be in the public domain so the security and cryptography can be verified, and not deliberately restricted for commercial or other reasons.
SQRL has some design-inherent and intentionalphishingdefenses,[6]but it is mainly intended to be for authentication, not anti-phishing, despite having some anti-phishing properties.[7]
For the protocol to be used on a website, two components are necessary: animplementation, that is part of theweb serviceto which the implementation authenticates, which displays aQR codeor specially craftedURLaccording to thespecificationsof the protocol, and abrowser pluginor amobile application, which can read this code in order to provide secure authentication.
The SQRL client usesone-way functionsand the user's single master password to decrypt a secret master key, from which it generates – in combination with the site domain name and optionally an additional sub-site identifier: e.g.,example.com, orexample.edu/chessclub– a (sub-)site-specificpublic/private key pair. It signs thetransaction tokenswith the private key and gives the public key to the site, so it can verify the encrypted data.
There are no "shared secrets" which a compromise of the site could expose to allow attacks on accounts at other sites. The only thing a successful attacker could get, the public key, would be limited to verifying signatures that are only used at the same site. Even though the user unlocks the master key with a single password, it never leaves the SQRL client; the individual sites do not receive any information from the SQRL process that could be used at any other site.
A number ofproof-of-conceptimplementations have been made for various platforms.
There are also various server-end test and debugging sites available.[22][23]
Steve Gibson states that SQRL is "open and free, as it should be", and that the solution is "unencumbered by patents".[2]After SQRL brought a lot of attention to QR-code-based authentication mechanisms, the suggested protocol was said by blogger Michael Beiter to have been patented earlier and thus not generally available for royalty-free use.[24][non-primary source needed]The patent in question (not expiring until 2030) was applied for by and granted to Spanish company GMV Soluciones Globales Internet SA (a division of the Madrid-based technology and aerospace corporationGMV Innovating Solutions), between 2008 and 2012 by the patent offices of the United States, the European Union, Spain, and Portugal.[25]
Gibson responded: "What those guys are doing as described in that patent is completely different from the way SQRL operates, so there would be no conflict between SQRL and their patent. Superficially, anything that uses a 2D code for authentication seems 'similar' ... and superficially all such solutions are. But the details matter, and the way SQRL operates is entirely different in the details."[26]
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WS-Federation(Web Services Federation) is anIdentity Federationspecification, developed by a group of companies:BEA Systems,BMC Software,CA Inc.(along with Layer 7 Technologies now a part of CA Inc.),IBM,Microsoft,Novell,Hewlett Packard Enterprise, andVeriSign. Part of the largerWeb Services Securityframework, WS-Federation defines mechanisms for allowing different security realms to broker information on identities, identity attributes and authentication.
The following draft specifications are associated withWS-Security:
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https://en.wikipedia.org/wiki/WS-Federation
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Inchaos theory, thebutterfly effectis the sensitive dependence oninitial conditionsin which a small change in one state of adeterministicnonlinear systemcan result in large differences in a later state.
The term is closely associated with the work of the mathematician and meteorologistEdward Norton Lorenz. He noted that the butterfly effect is derived from the example of the details of atornado(the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distantbutterflyflapping its wings several weeks earlier. Lorenz originally used a seagull causing a storm but was persuaded to make it more poetic with the use of a butterfly and tornado by 1972.[1][2]He discovered the effect when he observed runs of hisweather modelwith initial condition data that were rounded in a seemingly inconsequential manner. He noted that the weather model would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome.[3]
The idea that small causes may have large effects in weather was earlier acknowledged by the French mathematician and physicistHenri Poincaré. The American mathematician and philosopherNorbert Wieneralso contributed to this theory. Lorenz's work placed the concept ofinstabilityof theEarth's atmosphereonto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoingnonlinear dynamicsanddeterministic chaos.[4]
The concept of the butterfly effect has since been used outside the context of weather science as a broad term for any situation where a small change is supposed to be the cause of larger consequences.
InThe Vocation of Man(1800),Johann Gottlieb Fichtesays "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole".
Chaos theoryand the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidenced by the case of thethree-body problemby Poincaré in 1890.[5]He later proposed that such phenomena could be common, for example, in meteorology.[6]
In 1898,Jacques Hadamardnoted general divergence of trajectories in spaces of negative curvature.Pierre Duhemdiscussed the possible general significance of this in 1908.[5]
In 1950,Alan Turingnoted: "The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping."[7]
The idea that the death of one butterfly could eventually have a far-reachingripple effecton subsequent historical events made its earliest known appearance in "A Sound of Thunder", a 1952 short story byRay Bradburyin which a time traveller alters the future by inadvertently treading on a butterfly in the past.[8]
More precisely, though, almost the exact idea and the exact phrasing —of a tiny insect's wing affecting the entire atmosphere's winds— was published in a children's book which became extremely successful and well-known globally in 1962, the year before Lorenz published:
"...whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world."
-- The Princess of Pure Reason
In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.[9]
Lorenz wrote:
At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weakvacuum tubeor some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last [decimal] place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution.
In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper calledDeterministic Nonperiodic Flow[3][11](the calculations were performed on aRoyal McBeeLGP-30computer).[12][13]Elsewhere he stated:
One meteorologist remarked that if the theory were correct, one flap of asea gull'swings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.[13]
Following proposals from colleagues, in later speeches and papers, Lorenz used the more poeticbutterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of theAmerican Association for the Advancement of Sciencein 1972, Philip Merilees concoctedDoes the flap of a butterfly's wings in Brazil set off a tornado in Texas?as a title.[1]Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely.[14]
The phrase refers to the effect of a butterfly's wings creating tiny changes in theatmospherethat may ultimately alter the path of atornadoor delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings cancausethe tornado: in the sense that the flap of the wings is a part of the initial conditions of an interconnected complex web; one set of conditions leads to a tornado, while the other set of conditions doesn't. The flapping wing creates a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare:domino effect). Had the butterfly not flapped its wings, thetrajectoryof the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado.
The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development ofensemble forecasting, in which a number of forecasts are made from perturbed initial conditions.[15]
Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed.[16]David Orrellargues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.[17][18]Stephen Wolframalso notes that theLorenz equationsare highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations.[19]Recent studies using generalizedLorenz modelsthat included additional dissipative terms and nonlinearity suggested that a larger heating parameter is required for the onset of chaos.[20]
While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied[1]to work he published in 1969[21]which took the idea a step further. Lorenz proposed a mathematical model for how tiny motions in the atmosphere scale up to affect larger systems. He found that the systems in that model could only be predicted up to a specific point in the future, and beyond that, reducing the error in the initial conditions would not increase the predictability (as long as the error is not zero). This demonstrated that a deterministic system could be "observationally indistinguishable" from a non-deterministic one in terms of predictability. Recent re-examinations of this paper suggest that it offered a significant challenge to the idea that our universe is deterministic, comparable to the challenges offered by quantum physics.[22][23]
In the book entitledThe Essence of Chaospublished in 1993,[24]Lorenz defined butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." This feature is the same as sensitive dependence of solutions on initial conditions (SDIC) in .[3]In the same book, Lorenz applied the activity of skiing and developed an idealized skiing model for revealing the sensitivity of time-varying paths to initial positions. A predictability horizon is determined before the onset of SDIC.[25]
Recurrence, the approximate return of a system toward its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of makingcomplex systems, such as theweather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.
Adynamical systemdisplays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical. Lorenz[24]defined sensitive dependence as follows:
The property characterizing an orbit (i.e., a solution) if most other orbits that pass close to it at some point do not remain close to it as time advances.
IfMis thestate spacefor the mapft{\displaystyle f^{t}}, thenft{\displaystyle f^{t}}displays sensitive dependence to initial conditions if for any x inMand any δ > 0, there are y inM, with distanced(. , .)such that0<d(x,y)<δ{\displaystyle 0<d(x,y)<\delta }and such that
for some positive parametera. The definition does not require that all points from a neighborhood separate from the base pointx, but it requires one positiveLyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems.[26]
The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of thelogistic map:
which, unlike most chaotic maps, has aclosed-form solution:
where theinitial conditionparameterθ{\displaystyle \theta }is given byθ=1πsin−1(x01/2){\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}. For rationalθ{\displaystyle \theta }, after a finite number ofiterationsxn{\displaystyle x_{n}}maps into aperiodic sequence. Butalmost allθ{\displaystyle \theta }are irrational, and, for irrationalθ{\displaystyle \theta },xn{\displaystyle x_{n}}never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2nshows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keepsxn{\displaystyle x_{n}}folded within the range [0, 1].
The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example. The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive to initial conditions. They add the caveat: "Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat wrong."[27]
The concept of the butterfly effect encompasses several phenomena. The two kinds of butterfly effects, including the sensitive dependence on initial conditions,[3]and the ability of a tiny perturbation to create an organized circulation at large distances,[1]are not exactly the same.[28]In Palmer et al.,[22]a new type of butterfly effect is introduced, highlighting the potential impact of small-scale processes on finite predictability within the Lorenz 1969 model. Additionally, the identification of ill-conditioned aspects of the Lorenz 1969 model points to a practical form of finite predictability.[25]These two distinct mechanisms suggesting finite predictability in the Lorenz 1969 model are collectively referred to as the third kind of butterfly effect.[29]The authors in[29]have considered Palmer et al.'s suggestions and have aimed to present their perspective without raising specific contentions.
The third kind of butterfly effect with finite predictability, as discussed in,[22]was primarily proposed based on a convergent geometric series, known as Lorenz's and Lilly's formulas. Ongoing discussions are addressing the validity of these two formulas for estimating predictability limits in.[30]
A comparison of the two kinds of butterfly effects[1][3]and the third kind of butterfly effect[21][22][23]has been documented.[29]In recent studies,[25][31]it was reported that both meteorological and non-meteorological linear models have shown that instability plays a role in producing a butterfly effect, which is characterized by brief but significant exponential growth resulting from a small disturbance.
The first kind of butterfly effect (BE1), known as SDIC (Sensitive Dependence on Initial Conditions), is widely recognized and demonstrated through idealized chaotic models. However, opinions differ regarding the second kind of butterfly effect, specifically the impact of a butterfly flapping its wings on tornado formation, as indicated in two 2024 articles.[32][33]In more recent discussions published byPhysics Today,[34][35]it is acknowledged that the second kind of butterfly effect (BE2) has never been rigorously verified using a realistic weather model. While the studies suggest that BE2 is unlikely in the real atmosphere,[32][34]its invalidity in this context does not negate the applicability of BE1 in other areas, such as pandemics or historical events.[36]
For the third kind of butterfly effect, the limited predictability within the Lorenz 1969 model is explained by scale interactions in one article[22]and by system ill-conditioning in another more recent study.[25]
According to Lighthill (1986),[37]the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review,[38]it was found that Lorenz's perspective on the predictability limit can be condensed into the following statement:
Recently, a short video has been created to present Lorenz's perspective on predictability limit.[41]
A recent study refers to the two-week predictability limit, initially calculated in the 1960s with the Mintz-Arakawa model's five-day doubling time, as the "Predictability Limit Hypothesis."[42]Inspired by Moore's Law, this term acknowledges the collaborative contributions of Lorenz, Mintz, and Arakawa under Charney's leadership. The hypothesis supports the investigation into extended-range predictions using both partial differential equation (PDE)-based physics methods and Artificial Intelligence (AI) techniques.
By revealing coexisting chaotic and non-chaotic attractors within Lorenz models, Shen and his colleagues proposed a revised view that "weather possesses chaos and order", in contrast to the conventional view of "weather is chaotic".[43][44][45]As a result, sensitive dependence on initial conditions (SDIC) does not always appear. Namely, SDIC appears when two orbits (i.e., solutions) become the chaotic attractor; it does not appear when two orbits move toward the same point attractor. The above animation fordouble pendulummotion provides an analogy. For large angles of swing the motion of the pendulum is often chaotic.[46][47]By comparison, for small angles of swing, motions are non-chaotic.
Multistability is defined when a system (e.g., thedouble pendulumsystem) contains more than one bounded attractor that depends only on initial conditions. The multistability was illustrated using kayaking in Figure on the right side (i.e., Figure 1 of[48]) where the appearance of strong currents and a stagnant area suggests instability and local stability, respectively. As a result, when two kayaks move along strong currents, their paths display SDIC. On the other hand, when two kayaks move into a stagnant area, they become trapped, showing no typical SDIC (although a chaotic transient may occur). Such features of SDIC or no SDIC suggest two types of solutions and illustrate the nature of multistability.
By taking into consideration time-varying multistability that is associated with the modulation of large-scale processes (e.g., seasonal forcing) and aggregated feedback of small-scale processes (e.g., convection), the above revised view is refined as follows:
"The atmosphere possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons."[48][49]
The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases insemiclassicalandquantum physics, including atoms in strong fields and the anisotropicKepler problem.[50][51]Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments;[52][53]however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed byMartin Gutzwiller[54]and John B. Delos and co-workers.[55]The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist.[56]
Other authors suggest that the butterfly effect can be observed in quantum systems. Zbyszek P. Karkuszewski et al. consider the time evolution of quantum systems which have slightly differentHamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians.[57]David Poulin et al. presented a quantum algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected to slightly different dynamics". They consider fidelity decay to be "the closest quantum analog to the (purely classical) butterfly effect".[58]Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a givenHamiltonian system, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity.[59][60]This quantum butterfly effect has been demonstrated experimentally.[61]Quantum and semiclassical treatments of system sensitivity to initial conditions are known asquantum chaos.[52][59]
The butterfly effect has appeared across mediums such as literature (for instance,A Sound of Thunder), films and television (such asThe Simpsons), video games (such asLife Is Strange), webcomics (such asHomestuck), AI-driven expansive language models, and more.
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Incryptography,confusionanddiffusionare two properties of a securecipheridentified byClaude Shannonin his 1945 classified reportA Mathematical Theory of Cryptography.[1]These properties, when present, work together to thwart the application ofstatistics, and other methods ofcryptanalysis.
Confusion in asymmetric cipheris obscuring the local correlation between the input (plaintext), and output (ciphertext) by varying the application of thekeyto the data, while diffusion is hiding the plaintext statistics by spreading it over a larger area of ciphertext.[2]Although ciphers can be confusion-only (substitution cipher,one-time pad) or diffusion-only (transposition cipher), any "reasonable"block cipheruses both confusion and diffusion.[2]These concepts are also important in the design ofcryptographic hash functions, andpseudorandom number generators, where decorrelation of the generated values is the main feature. Diffusion (and itsavalanche effect) is also applicable tonon-cryptographic hash functions.
Confusion means that each binary digit (bit) of the ciphertext should depend on several parts of the key, obscuring the connections between the two.[3]
The property of confusion hides the relationship between the ciphertext and the key.
This property makes it difficult to find the key from the ciphertext and if a single bit in a key is changed, the calculation of most or all of the bits in the ciphertext will be affected.
Confusion increases the ambiguity of ciphertext and it is used by both block and stream ciphers.
Insubstitution–permutation networks, confusion is provided bysubstitution boxes.[4]
Diffusion means that if we change a single bit of the plaintext, then about half of the bits in the ciphertext should change, and similarly, if we change one bit of the ciphertext, then about half of the plaintext bits should change.[5]This is equivalent to the expectation that encryption schemes exhibit anavalanche effect.
The purpose of diffusion is to hide the statistical relationship between the ciphertext and the plain text. For example, diffusion ensures that any patterns in the plaintext, such as redundant bits, are not apparent in the ciphertext.[3]Block ciphers achieve this by "diffusing" the information about the plaintext's structure across the rows and columns of the cipher.
In substitution–permutation networks, diffusion is provided bypermutation boxes(a.k.a. permutation layer[4]). In the beginning of the 21st century a consensus had appeared where the designers preferred the permutation layer to consist oflinear Boolean functions, although nonlinear functions can be used, too.[4]
In Shannon's original definitions,confusionrefers to making the relationship between theciphertextand thesymmetric keyas complex and involved as possible;diffusionrefers to dissipating the statistical structure ofplaintextover the bulk ofciphertext. This complexity is generally implemented through a well-defined and repeatable series ofsubstitutionsandpermutations. Substitution refers to the replacement of certain components (usually bits) with other components, following certain rules. Permutation refers to manipulation of the order of bits according to some algorithm. To be effective, any non-uniformity of plaintext bits needs to be redistributed across much larger structures in the ciphertext, making that non-uniformity much harder to detect.
In particular, for a randomly chosen input, if one flips thei-th bit, then the probability that thej-th output bit will change should be one half, for anyiandj—this is termed thestrict avalanche criterion. More generally, one may require that flipping a fixed set of bits should change each output bit with probability one half.
One aim of confusion is to make it very hard to find the key even if one has a large number of plaintext-ciphertext pairs produced with the same key. Therefore, each bit of the ciphertext should depend on the entire key, and in different ways on different bits of the key. In particular, changing one bit of the key should change the ciphertext completely.
Design of a modernblock cipheruses both confusion and diffusion,[2]with confusion changing data between the input and the output by applying a key-dependent non-linear transformation (linear calculations are easier to reverse and thus are easier to break).
Confusion inevitably involves some diffusion,[6]so a design with a very wide-inputS-boxcan provide the necessary diffusion properties,[citation needed]but will be very costly in implementation. Therefore, the practical ciphers utilize relatively small S-boxes, operating on small groups of bits ("bundles"[7]). For example, the design of AES has 8-bit S-boxes,Serpent− 4-bit,BaseKingand3-way− 3-bit.[8]Small S-boxes provide almost no diffusion, so the resources are spent on simpler diffusion transformations.[6]For example, thewide trail strategypopularized by theRijndaeldesign, involves a linear mixing transformation that provides high diffusion,[9]although the security proofs do not depend on the diffusion layer being linear.[10]
One of the most researched cipher structures uses thesubstitution-permutation network(SPN) where eachroundincludes a layer of local nonlinear permutations (S-boxes) for confusion and alinear diffusiontransformation (usually a multiplication by a matrix over afinite field).[11]Modern block ciphers mostly follow the confusion layer/diffusion layer model, with the efficiency of the diffusion layer estimated using the so-calledbranch number, a numerical parameter that can reach the values+1{\displaystyle s+1}forsinput bundles for the perfect diffusion transformation.[12]Since the transformations that have high branch numbers (and thus require a lot of bundles as inputs) are costly in implementation, the diffusion layer is sometimes (for example, in the AES) composed from two sublayers, "local diffusion" that processes subsets of the bundles in abricklayerfashion (each subset is transformed independently) and "dispersion" that makes the bits that were "close" (within one subset of bundles) to become "distant" (spread to different subsets and thus be locally diffused within these new subsets on the next round).[13]
TheAdvanced Encryption Standard(AES) has both excellent confusion and diffusion. Its confusion look-up tables are very non-linear and good at destroying patterns.[14]Its diffusion stage spreads every part of the input to every part of the output: changing one bit of input changes half the output bits on average. Both confusion and diffusion are repeated multiple times for each input to increase the amount of scrambling. The secret key is mixed in at every stage so that an attacker cannot precalculate what the cipher does.
None of this happens when a simple one-stage scramble is based on a key. Input patterns would flow straight through to the output. It might look random to the eye but analysis would find obvious patterns and the cipher could be broken.
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Incomputing, awordis anyprocessordesign's natural unit of data. A word is a fixed-sizeddatumhandled as a unit by theinstruction setor the hardware of the processor. The number ofbitsor digits[a]in a word (theword size,word width, orword length) is an important characteristic of any specific processor design orcomputer architecture.
The size of a word is reflected in many aspects of a computer's structure and operation; the majority of theregistersin a processor are usually word-sized and the largest datum that can be transferred to and from theworking memoryin a single operation is a word in many (not all) architectures. The largest possibleaddresssize, used to designate a location in memory, is typically a hardware word (here, "hardware word" means the full-sized natural word of the processor, as opposed to any other definition used).
Documentation for older computers with fixed word size commonly states memory sizes in words rather than bytes or characters. The documentation sometimes usesmetric prefixescorrectly, sometimes with rounding, e.g.,65 kilowords(kW) meaning for 65536 words, and sometimes uses them incorrectly, withkilowords(kW) meaning 1024 words (210) and megawords (MW) meaning 1,048,576 words (220). With standardization on 8-bit bytes and byte addressability, stating memory sizes in bytes, kilobytes, and megabytes with powers of 1024 rather than 1000 has become the norm, although there is some use of theIECbinary prefixes.
Several of the earliest computers (and a few modern as well) usebinary-coded decimalrather than plainbinary, typically having a word size of 10 or 12decimaldigits, and some earlydecimal computershave no fixed word length at all. Early binary systems tended to use word lengths that were some multiple of 6-bits, with the 36-bit word being especially common onmainframe computers. The introduction ofASCIIled to the move to systems with word lengths that were a multiple of 8-bits, with 16-bit machines being popular in the 1970s before the move to modern processors with 32 or 64 bits.[1]Special-purpose designs likedigital signal processors, may have any word length from 4 to 80 bits.[1]
The size of a word can sometimes differ from the expected due tobackward compatibilitywith earlier computers. If multiple compatible variations or a family of processors share a common architecture and instruction set but differ in their word sizes, their documentation and software may become notationally complex to accommodate the difference (seeSize familiesbelow).
Depending on how a computer is organized, word-size units may be used for:
When a computer architecture is designed, the choice of a word size is of substantial importance. There are design considerations which encourage particular bit-group sizes for particular uses (e.g. for addresses), and these considerations point to different sizes for different uses. However, considerations of economy in design strongly push for one size, or a very few sizes related by multiples or fractions (submultiples) to a primary size. That preferred size becomes the word size of the architecture.
Charactersize was in the past (pre-variable-sizedcharacter encoding) one of the influences on unit of address resolution and the choice of word size. Before the mid-1960s, characters were most often stored in six bits; this allowed no more than 64 characters, so the alphabet was limited to upper case. Since it is efficient in time and space to have the word size be a multiple of the character size, word sizes in this period were usually multiples of 6 bits (in binary machines). A common choice then was the36-bit word, which is also a good size for the numeric properties of a floating point format.
After the introduction of theIBMSystem/360design, which uses eight-bit characters and supports lower-case letters, the standard size of a character (or more accurately, abyte) becomes eight bits. Word sizes thereafter are naturally multiples of eight bits, with 16, 32, and 64 bits being commonly used.
Early machine designs included some that used what is often termed avariable word length. In this type of organization, an operand has no fixed length. Depending on the machine and the instruction, the length might be denoted by a count field, by a delimiting character, or by an additional bit called, e.g., flag, orword mark. Such machines often usebinary-coded decimalin 4-bit digits, or in 6-bit characters, for numbers. This class of machines includes theIBM 702,IBM 705,IBM 7080,IBM 7010,UNIVAC 1050,IBM 1401,IBM 1620, andRCA301.
Most of these machines work on one unit of memory at a time and since each instruction or datum is several units long, each instruction takes several cycles just to access memory. These machines are often quite slow because of this. For example, instruction fetches on anIBM 1620 Model Itake 8 cycles (160 μs) just to read the 12 digits of the instruction (theModel IIreduced this to 6 cycles, or 4 cycles if the instruction did not need both address fields). Instruction execution takes a variable number of cycles, depending on the size of the operands.
The memory model of an architecture is strongly influenced by the word size. In particular, the resolution of a memory address, that is, the smallest unit that can be designated by an address, has often been chosen to be the word. In this approach, theword-addressablemachine approach, address values which differ by one designate adjacent memory words. This is natural in machines which deal almost always in word (or multiple-word) units, and has the advantage of allowing instructions to use minimally sized fields to contain addresses, which can permit a smaller instruction size or a larger variety of instructions.
When byte processing is to be a significant part of the workload, it is usually more advantageous to use thebyte, rather than the word, as the unit of address resolution. Address values which differ by one designate adjacent bytes in memory. This allows an arbitrary character within a character string to be addressed straightforwardly. A word can still be addressed, but the address to be used requires a few more bits than the word-resolution alternative. The word size needs to be an integer multiple of the character size in this organization. This addressing approach was used in the IBM 360, and has been the most common approach in machines designed since then.
When the workload involves processing fields of different sizes, it can be advantageous to address to the bit. Machines with bit addressing may have some instructions that use a programmer-defined byte size and other instructions that operate on fixed data sizes. As an example, on theIBM 7030[4]("Stretch"), a floating point instruction can only address words while an integer arithmetic instruction can specify a field length of 1-64 bits, a byte size of 1-8 bits and an accumulator offset of 0-127 bits.
In abyte-addressablemachine with storage-to-storage (SS) instructions, there are typically move instructions to copy one or multiple bytes from one arbitrary location to another. In a byte-oriented (byte-addressable) machine without SS instructions, moving a single byte from one arbitrary location to another is typically:
Individual bytes can be accessed on a word-oriented machine in one of two ways. Bytes can be manipulated by a combination of shift and mask operations in registers. Moving a single byte from one arbitrary location to another may require the equivalent of the following:
Alternatively many word-oriented machines implement byte operations with instructions using specialbyte pointersin registers or memory. For example, thePDP-10byte pointer contained the size of the byte in bits (allowing different-sized bytes to be accessed), the bit position of the byte within the word, and the word address of the data. Instructions could automatically adjust the pointer to the next byte on, for example, load and deposit (store) operations.
Different amounts of memory are used to store data values with different degrees of precision. The commonly used sizes are usually apower of twomultiple of the unit of address resolution (byte or word). Converting the index of an item in an array into the memory address offset of the item then requires only ashiftoperation rather than a multiplication. In some cases this relationship can also avoid the use of division operations. As a result, most modern computer designs have word sizes (and other operand sizes) that are a power of two times the size of a byte.
As computer designs have grown more complex, the central importance of a single word size to an architecture has decreased. Although more capable hardware can use a wider variety of sizes of data, market forces exert pressure to maintainbackward compatibilitywhile extending processor capability. As a result, what might have been the central word size in a fresh design has to coexist as an alternative size to the original word size in a backward compatible design. The original word size remains available in future designs, forming the basis of a size family.
In the mid-1970s,DECdesigned theVAXto be a 32-bit successor of the 16-bitPDP-11. They usedwordfor a 16-bit quantity, whilelongwordreferred to a 32-bit quantity; this terminology is the same as the terminology used for the PDP-11. This was in contrast to earlier machines, where the natural unit of addressing memory would be called aword, while a quantity that is one half a word would be called ahalfword. In fitting with this scheme, a VAXquadwordis 64 bits. They continued this 16-bit word/32-bit longword/64-bit quadword terminology with the 64-bitAlpha.
Another example is thex86family, of which processors of three different word lengths (16-bit, later 32- and 64-bit) have been released, whilewordcontinues to designate a 16-bit quantity. As software is routinelyportedfrom one word-length to the next, someAPIsand documentation define or refer to an older (and thus shorter) word-length than the full word length on the CPU that software may be compiled for. Also, similar to how bytes are used for small numbers in many programs, a shorter word (16 or 32 bits) may be used in contexts where the range of a wider word is not needed (especially where this can save considerable stack space or cache memory space). For example, Microsoft'sWindows APImaintains theprogramming languagedefinition ofWORDas 16 bits, despite the fact that the API may be used on a 32- or 64-bit x86 processor, where the standard word size would be 32 or 64 bits, respectively. Data structures containing such different sized words refer to them as:
A similar phenomenon has developed inIntel'sx86assembly language– because of the support for various sizes (and backward compatibility) in the instruction set, some instruction mnemonics carry "d" or "q" identifiers denoting "double-", "quad-" or "double-quad-", which are in terms of the architecture's original 16-bit word size.
An example with a different word size is theIBMSystem/360family. In theSystem/360 architecture,System/370 architectureandSystem/390architecture, there are 8-bitbytes, 16-bithalfwords, 32-bitwords and 64-bitdoublewords. Thez/Architecture, which is the 64-bit member of that architecture family, continues to refer to 16-bithalfwords, 32-bitwords, and 64-bitdoublewords, and additionally features 128-bitquadwords.
In general, new processors must use the same data word lengths and virtual address widths as an older processor to havebinary compatibilitywith that older processor.
Often carefully written source code – written withsource-code compatibilityandsoftware portabilityin mind – can be recompiled to run on a variety of processors, even ones with different data word lengths or different address widths or both.
[8][9]
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Incomputer security,challenge-response authenticationis a family of protocols in which one party presents a question ("challenge") and another party must provide a valid answer ("response") to beauthenticated.[1]
The simplest example of a challenge-response protocol ispasswordauthentication, where the challenge is asking for the password and the valid response is the correct password.
Anadversarywho caneavesdropon a password authentication can authenticate themselves by reusing the intercepted password. One solution is to issue multiple passwords, each of them marked with an identifier. The verifier can then present an identifier, and the prover must respond with the correct password for that identifier. Assuming that the passwords are chosen independently, an adversary who intercepts one challenge-response message pair has no clues to help with a different challenge at a different time.
For example, when othercommunications securitymethods are unavailable, theU.S. militaryuses theAKAC-1553TRIAD numeral cipher to authenticate and encrypt some communications. TRIAD includes a list of three-letter challenge codes, which the verifier is supposed to choose randomly from, and random three-letter responses to them. For added security, each set of codes is only valid for a particular time period which is ordinarily 24 hours.
Another basic challenge-response technique works as follows.Bobis controlling access to some resource, and Alice is seeking entry. Bob issues the challenge "52w72y". Alice must respond with the one string of characters which "fits" the challenge Bob issued. The "fit" is determined by an algorithm defined in advance, and known by both Bob and Alice. The correct response might be as simple as "63x83z", with the algorithm changing each character of the challenge using aCaesar cipher. In reality, the algorithm would be much more complex. Bob issues a different challenge each time, and thus knowing a previous correct response (even if it is not obfuscated by the means of communication) does not allow an adversary to determine the current correct response.
Challenge-response protocols are also used in non-cryptographic applications.CAPTCHAs, for example, are meant to allow websites and applications to determine whether an interaction was performed by a genuine user rather than aweb scraperorbot. In early CAPTCHAs, the challenge sent to the user was a distorted image of some text, and the user responded by transcribing the text. The distortion was designed to make automatedoptical character recognition(OCR) difficult and prevent a computer program from passing as a human.
Non-cryptographic authentication was generally adequate in the days before theInternet, when the user could be sure that the system asking for the password was really the system they were trying to access, and that nobody was likely to be eavesdropping on thecommunication channel. To address the insecure channel problem, a more sophisticated approach is necessary. Many cryptographic solutions involvetwo-way authentication;both the user and the system must verify that they know theshared secret(the password), without the secret ever being transmittedin the clearover the communication channel.
One way this is done involves using the password as theencryptionkey to transmit some randomly generated information as thechallenge, whereupon the other end must return as itsresponsea similarly encrypted value which is some predetermined function of the originally offered information, thus proving that it was able to decrypt the challenge. For instance, inKerberos, the challenge is an encrypted integerN, while the response is the encrypted integerN + 1, proving that the other end was able to decrypt the integerN. A hash function can also be applied to a password and a random challenge value to create a response value. Another variation uses a probabilistic model to provide randomized challenges conditioned on model input.[2]
Such encrypted or hashed exchanges do not directly reveal the password to an eavesdropper. However, they may supply enough information to allow an eavesdropper to deduce what the password is, using adictionary attackorbrute-force attack. The use of information which is randomly generated on each exchange (and where the response is different from the challenge) guards against the possibility of areplay attack, where a malicious intermediary simply records the exchanged data and retransmits it at a later time to fool one end into thinking it has authenticated a new connection attempt from the other.
Authentication protocols usually employ acryptographic nonceas the challenge to ensure that every challenge-response sequence is unique. This protects againstEavesdroppingwith a subsequentreplay attack. If it is impractical to implement a true nonce, a strongcryptographically secure pseudorandom number generatorandcryptographic hash functioncan generate challenges that are highly unlikely to occur more than once. It is sometimes important not to use time-based nonces, as these can weaken servers in different time zones and servers with inaccurate clocks. It can also be important to use time-based nonces and synchronized clocks if the application is vulnerable to a delayed message attack. This attack occurs where an attacker copies a transmission whilst blocking it from reaching the destination, allowing them to replay the captured transmission after a delay of their choosing. This is easily accomplished on wireless channels. The time-based nonce can be used to limit the attacker to resending the message but restricted by an expiry time of perhaps less than one second, likely having no effect upon the application and so mitigating the attack.
Mutual authenticationis performed using a challenge-response handshake in both directions; the server ensures that the client knows the secret, and the clientalsoensures that the server knows the secret, which protects against a rogue server impersonating the real server.
Challenge-response authentication can help solve the problem of exchanging session keys for encryption. Using akey derivation function, the challenge value and the secret may be combined to generate an unpredictable encryption key for the session. This is particularly effective against a man-in-the-middle attack, because the attacker will not be able to derive the session key from the challenge without knowing the secret, and therefore will not be able to decrypt the data stream.
where
This particular example is vulnerable to areflection attack.
To avoid storage of passwords, some operating systems (e.g.Unix-type) store ahash of the passwordrather than storing the password itself. During authentication, the system need only verify that the hash of the password entered matches the hash stored in the password database. This makes it more difficult for an intruder to get the passwords, since the password itself is not stored, and it is very difficult to determine a password that matches a given hash. However, this presents a problem for many (but not all) challenge-response algorithms, which require both the client and the server to have a shared secret. Since the password itself is not stored, a challenge-response algorithm will usually have to use the hash of the password as the secret instead of the password itself. In this case, an intruder can use the actual hash, rather than the password, which makes the stored hashes just as sensitive as the actual passwords.SCRAMis a challenge-response algorithm that avoids this problem.
Examples of more sophisticated challenge-responsealgorithmsare:
Some people consider aCAPTCHAa kind of challenge-response authentication that blocksspambots.[4]
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Incomputer science, ahash listis typically alistofhashesof the data blocks in a file or set of files. Lists of hashes are used for many different purposes, such as fast table lookup (hash tables) and distributed databases (distributed hash tables).
A hash list is an extension of the concept of hashing an item (for instance, a file). A hash list is asubtreeof aMerkle tree.
Often, an additional hash of the hash list itself (atop hash, also calledroot hashormaster hash) is used. Before downloading a file on ap2p network, in most cases the top hash is acquired from a trusted source, for instance a friend or a web site that is known to have good recommendations of files to download. When the top hash is available, the hash list can be received from any non-trusted source, like any peer in the p2p network. Then the received hash list is checked against the trusted top hash, and if the hash list is damaged or fake, another hash list from another source will be tried until the program finds one that matches the top hash.
In some systems (for example,BitTorrent), instead of a top hash the whole hash list is available on a web site in a small file. Such a "torrent file" contains a description, file names, a hash list and some additional data.
Hash lists can be used to protect any kind of data stored, handled and transferred in and between computers. An important use of hash lists is to make sure that data blocks received from other peers in apeer-to-peer networkare received undamaged and unaltered, and to check that the other peers do not "lie" and send fake blocks.[citation needed]
Usually acryptographic hash functionsuch asSHA-256is used for the hashing. If the hash list only needs to protect against unintentional damage unsecuredchecksumssuch asCRCscan be used.[citation needed]
Hash lists are better than a simple hash of the entire file since, in the case of a data block being damaged, this is noticed, and only the damaged block needs to be redownloaded. With only a hash of the file, many undamaged blocks would have to be redownloaded, and the file reconstructed and tested until the correct hash of the entire file is obtained. Hash lists also protect against nodes that try to sabotage by sending fake blocks, since in such a case the damaged block can be acquired from some other source.[citation needed]
Hash lists are used to identifyCSAMonline.[1]
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Linked timestampingis a type oftrusted timestampingwhere issued time-stamps are related to each other. Each time-stamp would contain data that authenticates the time-stamp before it, the authentication would be authenticating the entire message, including the previous time-stamps authentication, making a chain. This makes it impossible to add a time-stamp in to the middle of the chain, as any time-stamps afterwards would be different.
Linked timestamping creates time-stamp tokens which are dependent on each other, entangled in someauthenticateddata structure. Later modification of the issued time-stamps would invalidate this structure. The temporal order of issued time-stamps is also protected by this data structure, making backdating of the issued time-stamps impossible, even by the issuing server itself.
The top of the authenticated data structure is generallypublishedin some hard-to-modify and widely witnessed media, like printednewspaperor publicblockchain. There are no (long-term)private keysin use, avoidingPKI-related risks.
Suitable candidates for the authenticated data structure include:
The simplest linear hash chain-based time-stamping scheme is illustrated in the following diagram:
The linking-basedtime-stamping authority(TSA) usually performs the following distinct functions:
Linked timestamping is inherently more secure than the usual, public-key signature based time-stamping. All consequential time-stamps "seal" previously issued ones - hash chain (or other authenticated dictionary in use) could be built only in one way; modifying issued time-stamps is nearly as hard as finding a preimage for the usedcryptographic hash function. Continuity of operation is observable by users; periodic publications in widely witnessed media provide extra transparency.
Tampering with absolute time values could be detected by users, whose time-stamps are relatively comparable by system design.
Absence of secret keys increases system trustworthiness. There are no keys to leak and hash algorithms are considered more future-proof[1]than modular arithmetic based algorithms, e.g.RSA.
Linked timestamping scales well - hashing is much faster than public key cryptography. There is no need for specific cryptographic hardware with its limitations.
The common technology[2]for guaranteeing long-term attestation value of the issued time-stamps (and digitally signed data[3]) is periodic over-time-stamping of the time-stamp token. Because of missing key-related risks and of the plausible safety margin of the reasonably chosen hash function this over-time-stamping period of hash-linked token could be an order of magnitude longer than of public-key signed token.
Stuart HaberandW. Scott Stornettaproposed[4]in 1990 to link issued time-stamps together into linear hash-chain, using acollision-resistanthash function. The main rationale was to diminishTSAtrust requirements.
Tree-like schemes and operating in rounds were proposed by Benaloh and de Mare in 1991[5]and by Bayer, Haber and Stornetta in 1992.[6]
Benaloh and de Mare constructed a one-way accumulator[7]in 1994 and proposed its use in time-stamping. When used for aggregation, one-way accumulator requires only one constant-time computation for round membership verification.
Surety[8]started the first commercial linked timestamping service in January 1995. Linking scheme is described and its security is analyzed in the following article[9]by Haber and Sornetta.
Buldaset al. continued with further optimization[10]and formal analysis of binary tree and threaded tree[11]based schemes.
Skip-list based time-stamping system was implemented in 2005; related algorithms are quite efficient.[12]
Security proof for hash-function based time-stamping schemes was presented by Buldas, Saarepera[13]in 2004. There is an explicit upper boundN{\displaystyle N}for the number of time stamps issued during the aggregation period; it is suggested that it is probably impossible to prove the security without this explicit bound - the so-called black-box reductions will fail in this task. Considering that all known practically relevant and efficient security proofs are black-box, this negative result is quite strong.
Next, in 2005 it was shown[14]that bounded time-stamping schemes with a trusted audit party (who periodically reviews the list of all time-stamps issued during an aggregation period) can be madeuniversally composable- they remain secure in arbitrary environments (compositions with other protocols and other instances of the time-stamping protocol itself).
Buldas, Laur showed[15]in 2007 that bounded time-stamping schemes are secure in a very strong sense - they satisfy the so-called "knowledge-binding" condition. The security guarantee offered by Buldas, Saarepera in 2004 is improved by diminishing the security loss coefficient fromN{\displaystyle N}toN{\displaystyle {\sqrt {N}}}.
The hash functions used in the secure time-stamping schemes do not necessarily have to be collision-resistant[16]or even one-way;[17]secure time-stamping schemes are probably possible even in the presence of a universal collision-finding algorithm (i.e. universal and attacking program that is able to findcollisionsfor any hash function). This suggests that it is possible to find even stronger proofs based on some other properties of the hash functions.
At the illustration above hash tree based time-stamping system works in rounds (t{\displaystyle t},t+1{\displaystyle t+1},t+2{\displaystyle t+2}, ...), with one aggregation tree per round. Capacity of the system (N{\displaystyle N}) is determined by the tree size (N=2l{\displaystyle N=2^{l}}, wherel{\displaystyle l}denotes binary tree depth). Current security proofs work on the assumption that there is a hard limit of the aggregation tree size, possibly enforced by the subtree length restriction.
ISO 18014part 3 covers 'Mechanisms producing linked tokens'.
American National Standardfor Financial Services, "Trusted Timestamp Management and Security" (ANSI ASC X9.95 Standard) from June 2005 covers linking-based and hybrid time-stamping schemes.
There is noIETFRFCor standard draft about linking based time-stamping.RFC4998(Evidence Record Syntax) encompasses hash tree and time-stamp as an integrity guarantee for long-term archiving.
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This is a comprehensive list ofvolunteer computingprojects, which are a type ofdistributed computingwhere volunteers donate computing time to specific causes. The donated computing power comes from idleCPUsandGPUsinpersonal computers,video game consoles,[1]andAndroid devices.
Each project seeks to utilize the computing power of many internet connected devices to solve problems and perform tedious, repetitive research in a very cost effective manner.
2014-06-01[139]
2011-08-23[146]
2017-04-07[151]
2022-09[155]
2016[164]
2017[169]
2024-01
2018-04-20[180]
2014-06[183]
2014-05-23[187]
2012[197]
2004-03-08[198]
2011-06[200]
2001-09-03[202]
2010-02-21
2011-02[208]
2016-07[211]
2018-06-05[217]
2017-03[220]
2009-11-15[224]
2016-01-28[230]
2022-10-02[238]
2010
2016[248]
2013-02-16[251]
2 Spy Hill Research[254]
Tested BOINC's forum software for possible use byInteractions in Understanding the Universe[256]
2016-10-04[258]
2020-05-31[264]
2013-01
2023-06-01[270]
2022-09-29
Merged with PrimeGrid.
2012-08
2017-02[281]
2012-08[284]
2020-03-31[289]
2020-03-31[289]
2014-05-30
2011-09-28[297]
2018-01[300]
2013-09-05
2018-05-02[306]
2010-11[311]
2019-03[314]
2019-03[314]
2017[319]
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Incryptography, abrute-force attackconsists of an attacker submitting manypasswordsorpassphraseswith the hope of eventually guessing correctly. The attacker systematically checks all possible passwords and passphrases until the correct one is found. Alternatively, the attacker can attempt to guess thekeywhich is typically created from the password using akey derivation function. This is known as anexhaustive key search. This approach doesn't depend on intellectual tactics; rather, it relies on making several attempts.[citation needed]
A brute-force attack is acryptanalytic attackthat can, in theory, be used to attempt to decrypt any encrypted data (except for data encrypted in aninformation-theoretically securemanner).[1]Such an attack might be used when it is not possible to take advantage of other weaknesses in an encryption system (if any exist) that would make the task easier.
When password-guessing, this method is very fast when used to check all short passwords, but for longer passwords other methods such as thedictionary attackare used because a brute-force search takes too long. Longer passwords, passphrases and keys have more possible values, making them exponentially more difficult to crack than shorter ones due to diversity of characters.[2]
Brute-force attacks can be made less effective byobfuscatingthe data to be encoded making it more difficult for an attacker to recognize when the code has been cracked or by making the attacker do more work to test each guess. One of the measures of the strength of an encryption system is how long it would theoretically take an attacker to mount a successful brute-force attack against it.[3]
Brute-force attacks are an application of brute-force search, the general problem-solving technique of enumerating all candidates and checking each one. The word 'hammering' is sometimes used to describe a brute-force attack,[4]with 'anti-hammering' for countermeasures.[5]
Brute-force attacks work by calculating every possible combination that could make up a password and testing it to see if it is the correct password. As the password's length increases, the amount of time, on average, to find the correct password increases exponentially.[6]
The resources required for a brute-force attack growexponentiallywith increasingkey size, not linearly. Although U.S. export regulations historically restricted key lengths to 56-bitsymmetric keys(e.g.Data Encryption Standard), these restrictions are no longer in place, so modern symmetric algorithms typically use computationally stronger 128- to 256-bit keys.
There is a physical argument that a 128-bit symmetric key is computationally secure against brute-force attack. TheLandauer limitimplied by the laws of physics sets a lower limit on the energy required to perform a computation ofkT·ln 2per bit erased in a computation, whereTis the temperature of the computing device inkelvins,kis theBoltzmann constant, and thenatural logarithmof 2 is about 0.693 (0.6931471805599453). No irreversible computing device can use less energy than this, even in principle.[7]Thus, in order to simply flip through the possible values for a 128-bit symmetric key (ignoring doing the actual computing to check it) would, theoretically, require2128− 1bit flips on a conventional processor. If it is assumed that the calculation occurs near room temperature (≈300 K), the Von Neumann-Landauer Limit can be applied to estimate the energy required as ≈1018joules, which is equivalent to consuming 30gigawattsof power for one year. This is equal to 30×109W×365×24×3600 s = 9.46×1017J or 262.7 TWh (about 0.1% of theyearly world energy production). The full actual computation – checking each key to see if a solution has been found – would consume many times this amount. Furthermore, this is simply the energy requirement for cycling through the key space; the actual time it takes to flip each bit is not considered, which is certainly greater than 0 (seeBremermann's limit).[citation needed]
However, this argument assumes that the register values are changed using conventional set and clear operations, which inevitably generateentropy. It has been shown that computational hardware can be designed not to encounter this theoretical obstruction (seereversible computing), though no such computers are known to have been constructed.[citation needed]
As commercial successors of governmentalASICsolutions have become available, also known ascustom hardware attacks, two emerging technologies have proven their capability in the brute-force attack of certain ciphers. One is moderngraphics processing unit(GPU) technology,[8][page needed]the other is thefield-programmable gate array(FPGA) technology. GPUs benefit from their wide availability and price-performance benefit, FPGAs from theirenergy efficiencyper cryptographic operation. Both technologies try to transport the benefits of parallel processing to brute-force attacks. In case of GPUs some hundreds, in the case of FPGA some thousand processing units making them much better suited to cracking passwords than conventional processors. For instance in 2022, 8Nvidia RTX 4090GPU were linked together to test password strength by using the softwareHashcatwith results that showed 200 billion eight-characterNTLMpassword combinations could be cycled through in 48 minutes.[9][10]
Various publications in the fields of cryptographic analysis have proved the energy efficiency of today's FPGA technology, for example, the COPACOBANA FPGA Cluster computer consumes the same energy as a single PC (600 W), but performs like 2,500 PCs for certain algorithms. A number of firms provide hardware-based FPGA cryptographic analysis solutions from a single FPGAPCI Expresscard up to dedicated FPGA computers.[citation needed]WPAandWPA2encryption have successfully been brute-force attacked by reducing the workload by a factor of 50 in comparison to conventional CPUs[11][12]and some hundred in case of FPGAs.
Advanced Encryption Standard(AES) permits the use of 256-bit keys. Breaking a symmetric 256-bit key by brute-force requires 2128times more computational power than a 128-bit key. One of the fastest supercomputers in 2019 has a speed of 100petaFLOPSwhich could theoretically check 100 trillion (1014) AES keys per second (assuming 1000 operations per check), but would still require 3.67×1055years to exhaust the 256-bit key space.[13]
An underlying assumption of a brute-force attack is that the complete key space was used to generate keys, something that relies on an effectiverandom number generator, and that there are no defects in the algorithm or its implementation. For example, a number of systems that were originally thought to be impossible to crack by brute-force have nevertheless beencrackedbecause thekey spaceto search through was found to be much smaller than originally thought, because of a lack of entropy in theirpseudorandom number generators. These includeNetscape's implementation ofSecure Sockets Layer(SSL) (cracked byIan GoldbergandDavid Wagnerin 1995) and aDebian/Ubuntuedition ofOpenSSLdiscovered in 2008 to be flawed.[14][15]A similar lack of implemented entropy led to the breaking ofEnigma'scode.[16][17]
Credential recycling is thehackingpractice of re-using username and password combinations gathered in previous brute-force attacks. A special form of credential recycling ispass the hash, whereunsaltedhashed credentials are stolen and re-used without first being brute-forced.[18]
Certain types of encryption, by their mathematical properties, cannot be defeated by brute-force. An example of this isone-time padcryptography, where everycleartextbit has a corresponding key from a truly random sequence of key bits. A 140 character one-time-pad-encoded string subjected to a brute-force attack would eventually reveal every 140 character string possible, including the correct answer – but of all the answers given, there would be no way of knowing which was the correct one. Defeating such a system, as was done by theVenona project, generally relies not on pure cryptography, but upon mistakes in its implementation, such as the key pads not being truly random, intercepted keypads, or operators making mistakes.[19]
In case of anofflineattack where the attacker has gained access to the encrypted material, one can try key combinations without the risk of discovery or interference. In case ofonlineattacks, database and directory administrators can deploy countermeasures such as limiting the number of attempts that a password can be tried, introducing time delays between successive attempts, increasing the answer's complexity (e.g., requiring aCAPTCHAanswer or employingmulti-factor authentication), and/or locking accounts out after unsuccessful login attempts.[20][page needed]Website administrators may prevent a particular IP address from trying more than a predetermined number of password attempts against any account on the site.[21]Additionally, the MITRE D3FEND framework provides structured recommendations for defending against brute-force attacks by implementing strategies such as network traffic filtering, deploying decoy credentials, and invalidating authentication caches.[22]
In a reverse brute-force attack (also called password spraying), a single (usually common) password is tested against multiple usernames or encrypted files.[23]The process may be repeated for a select few passwords. In such a strategy, the attacker is not targeting a specific user.
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ECRYPT(European Network of Excellence in Cryptology) was a 4-yearEuropeanresearch initiative launched on 1 February 2004 with the stated objective of promoting the collaboration of European researchers ininformation security, and especially incryptologyanddigital watermarking.
ECRYPT listed five core research areas, termed "virtual laboratories":symmetric key algorithms(STVL),public key algorithms(AZTEC),protocol(PROVILAB), secure and efficient implementations (VAMPIRE) andwatermarking(WAVILA).
In August 2008 the network started another 4-year phase asECRYPT II.
During the project, algorithms and key lengths were evaluated yearly. The most recent of these documents is dated 30 September 2012.[1]
Considering the budget of a large intelligence agency to be about US$300 million for a singleASICmachine, the recommendedminimumkey size is 84 bits, which would give protection for a few months. In practice, most commonly used algorithms have key sizes of 128 bits or more, providing sufficient security also in the case that the chosen algorithm is slightly weakened by cryptanalysis.
Different kinds of keys are compared in the document (e.g. RSA keys vs.ECkeys). This "translation table" can be used to roughly equate keys of other types of algorithms with symmetric encryption algorithms. In short, 128 bit symmetric keys are said to be equivalent to 3248 bits RSA keys or 256-bit EC keys. Symmetric keys of 256 bits are roughly equivalent to 15424 bit RSA keys or 512 bit EC keys. Finally 2048 bit RSA keys are said to be equivalent to 103 bit symmetric keys.
Among key sizes, 8 security levels are defined, from the lowest "Attacks possible in real-time by individuals" (level 1, 32 bits) to "Good for the foreseeable future, also against quantum computers unlessShor's algorithmapplies" (level 8, 256 bits). For general long-term protection (30 years), 128 bit keys are recommended (level 7).
Many different primitives and algorithms are evaluated. The primitives are:
Note that the list of algorithms and schemes is non-exhaustive (the document contains more algorithms than are mentioned here).
This document, dated 11 January 2013, provides "an exhaustive overview of every computational assumption that has been used in public key cryptography."[2]
The "Vampire lab" produced over 80 peer-reviewed and joined authored publications during the four years of the project. This final document looks back on results and discusses newly arising research directions. The goals were to advance attacks and countermeasures; bridging the gap between cryptographic protocol designers and smart card implementers; and to investigate countermeasures against power analysis attacks (contact-based and contact-less).[3]
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Incryptography, asponge functionorsponge constructionis any of a class ofalgorithmswith finiteinternal statethat take an inputbit streamof any length and produce an output bit stream of any desired length. Sponge functions have both theoretical and practical uses. They can be used to model or implement manycryptographic primitives, includingcryptographic hashes,message authentication codes,mask generation functions,stream ciphers,pseudo-random number generators, andauthenticated encryption.[1]
A sponge function is built from three components:[2]
Sis divided into two sections: one of sizer(the bitrate) and the remaining part of sizec(the capacity). These sections are denotedRandCrespectively.
fproduces apseudorandom permutationof the2b{\displaystyle 2^{b}}states fromS.
Pappends enough bits to the input string so that the length of the padded input is a whole multiple of the bitrate,r. This means the input is segmented into blocks ofrbits.
The sponge function "absorbs" (in thespongemetaphor) all blocks of a padded input string as follows:
The sponge function output is now ready to be produced ("squeezed out") as follows:
If less thanrbits remain to be output, thenRwill be truncated (only part ofRwill be output).
Another metaphor describes the state memory as an "entropy pool", with input "poured into" the pool, and the transformation function referred to as "stirring the entropy pool".[3]
Note that input bits are never XORed into theCportion of the state memory, nor are any bits ofCever output directly. The extent to whichCis altered by the input depends entirely on the transformation functionf.In hash applications, resistance tocollisionorpreimage attacksdepends onC, and its size (the "capacity"c) is typically twice the desired resistance level.
It is also possible to absorb and squeeze in an alternating fashion.[1]This operation is called the duplex construction or duplexing. It can be the basis of a single pass authenticated encryption system. This have also been used as an efficient variant of theFiat-Shamir transformationfor some protocols.[4]
It is possible to omit the XOR operations during absorption, while still maintaining the chosensecurity level.[1]In this mode, in the absorbing phase, the next block of the input overwrites theRpart of the state. This allows keeping a smaller state between the steps. Since theRpart will be overwritten anyway, it can be discarded in advance, only theCpart must be kept.
Sponge functions have both theoretical and practical uses. In theoretical cryptanalysis, arandom sponge functionis a sponge construction wherefis a random permutation or transformation, as appropriate. Random sponge functions capture more of the practical limitations of cryptographic primitives than does the widely usedrandom oraclemodel, in particular the finite internal state.[5]
The sponge construction can also be used to build practical cryptographic primitives. For example, theKeccakcryptographic sponge with a 1600-bit state has been selected byNISTas the winner in theSHA-3 competition. The strength of Keccak derives from the intricate, multi-round permutationfthat its authors developed.[6]TheRC4-redesign calledSpritzrefers to the sponge-construct to define the algorithm.
For other examples, a sponge function can be used to buildauthenticated encryptionwith associated data (AEAD),[3]as well aspassword hashingschemes.[7]
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Multi-factor authentication(MFA;two-factor authentication, or2FA) is anelectronic authenticationmethod in which a user is granted access to awebsiteorapplicationonly after successfully presenting two or more distinct types of evidence (orfactors) to anauthenticationmechanism. MFA protectspersonal data—which may include personal identification orfinancial assets—from being accessed by an unauthorized third party that may have been able to discover, for example, a singlepassword.
Usage of MFA has increased in recent years. Security issues which can cause the bypass of MFA arefatigue attacks,phishingandSIM swapping.[1]
Accounts with MFA enabled are significantly less likely to be compromised.[2]
Authentication takes place when someone tries tolog intoa computer resource (such as acomputer network, device, or application). The resource requires the user to supply theidentityby which the user is known to the resource, along with evidence of the authenticity of the user's claim to that identity. Simple authentication requires only one such piece of evidence (factor), typically a password, or occasionally multiple pieces of evidence all of the same type, as with a credit card number and a card verification code (CVC). For additional security, the resource may require more than one factor—multi-factor authentication, or two-factor authentication in cases where exactly two types of evidence are to be supplied.[3]
The use of multiple authentication factors to prove one's identity is based on the premise that an unauthorized actor is unlikely to be able to supply all of the factors required for access. If, in an authentication attempt, at least one of the components is missing or supplied incorrectly, the user's identity is not established with sufficient certainty and access to the asset (e.g., a building, or data) being protected by multi-factor authentication then remains blocked. The authentication factors of a multi-factor authentication scheme may include:[4]
An example of two-factor authentication is the withdrawing of money from anATM; only the correct combination of a physically presentbank card(something the user possesses) and a PIN (something the user knows) allows the transaction to be carried out. Two other examples are to supplement a user-controlled password with aone-time password(OTP) or code generated or received by anauthenticator(e.g. asecurity tokenorsmartphone) that only the user possesses.[5]
Anauthenticatorapp enables two-factor authentication in a different way, by showing a randomly generated and constantly refreshing code, rather than sending anSMSor using another method.[6]This code is aTime-based one-time password(aTOTP)), and the authenticator app contains the key material that allows the generation of these codes.
Knowledge factors ("something only the user knows") are a form of authentication. In this form, the user is required to prove knowledge of a secret in order to authenticate.
A password is a secret word or string of characters that is used for user authentication. This is the most commonly used mechanism of authentication.[4]Many multi-factor authentication techniques rely on passwords as one factor of authentication. Variations include both longer ones formed from multiple words (apassphrase) and the shorter, purely numeric, PIN commonly used forATMaccess. Traditionally, passwords are expected to bememorized, but can also be written down on a hidden paper or text file.
Possession factors ("something only the user has") have been used for authentication for centuries, in the form of a key to a lock. The basic principle is that the key embodies a secret that is shared between the lock and the key, and the same principle underlies possession factor authentication in computer systems. Asecurity tokenis an example of a possession factor.
Disconnected tokenshave no connections to the client computer. They typically use a built-in screen to display the generated authentication data, which is manually typed in by the user. This type of token mostly uses aOTPthat can only be used for that specific session.[7]
Connected tokensaredevicesthat arephysicallyconnected to the computer to be used. Those devices transmit data automatically.[8]There are a number of different types, including USB tokens,smart cardsandwireless tags.[8]Increasingly,FIDO2capable tokens, supported by theFIDO Allianceand theWorld Wide Web Consortium(W3C), have become popular with mainstream browser support beginning in 2015.
Asoftware token(a.k.a.soft token) is a type of two-factor authentication security device that may be used to authorize the use of computer services. Software tokens are stored on a general-purpose electronic device such as adesktop computer,laptop,PDA, ormobile phoneand can be duplicated. (Contrasthardware tokens, where the credentials are stored on a dedicated hardware device and therefore cannot be duplicated, absent physical invasion of the device). A soft token may not be a device the user interacts with. Typically an X.509v3 certificate is loaded onto the device and stored securely to serve this purpose.[citation needed]
Multi-factor authenticationcan also be applied in physical security systems. These physical security systems are known and commonly referred to as access control. Multi-factor authentication is typically deployed in access control systems through the use, firstly, of a physical possession (such as a fob,keycard, orQR-codedisplayed on a device) which acts as the identification credential, and secondly, a validation of one's identity such as facial biometrics or retinal scan. This form of multi-factor authentication is commonly referred to as facial verification or facial authentication.
Inherent factors ("something the user is"), are factors associated with the user, and are usuallybiometricmethods, includingfingerprint,face,[9]voice, oririsrecognition. Behavioral biometrics such askeystroke dynamicscan also be used.
Increasingly, a fourth factor is coming into play involving the physical location of the user. While hard wired to the corporate network, a user could be allowed to login using only a pin code. Whereas if the user was off the network or working remotely, a more secure MFA method such as entering a code from a soft token as well could be required. Adapting the type of MFA method and frequency to a users' location will enable you to avoid risks common to remote working.[10]
Systems for network admission control work in similar ways where the level of network access can be contingent on the specific network a device is connected to, such asWi-Fivs wired connectivity. This also allows a user to move between offices and dynamically receivethe same level of network access[clarification needed]in each.[citation needed]
Two-factor authentication over text message was developed as early as 1996, when AT&T described a system for authorizing transactions based on an exchange of codes over two-way pagers.[11][12]
Many multi-factor authentication vendors offer mobile phone-based authentication. Some methods include push-based authentication,QR code-based authentication, one-time password authentication (event-based and time-based), and SMS-based verification. SMS-based verification suffers from some security concerns. Phones can be cloned, apps can run on several phones and cell-phone maintenance personnel can read SMS texts. Not least, cell phones can be compromised in general, meaning the phone is no longer something only the user has.
The major drawback of authentication including something the user possesses is that the user must carry around the physical token (the USB stick, the bank card, the key or similar), practically at all times. Loss and theft are risks. Many organizations forbid carrying USB and electronic devices in or out of premises owing tomalwareand data theft risks, and most important machines do not have USB ports for the same reason. Physical tokens usually do not scale, typically requiring a new token for each new account and system. Procuring and subsequently replacing tokens of this kind involves costs. In addition, there are inherent conflicts and unavoidable trade-offs between usability and security.[13]
Two-step authentication involvingmobile phonesandsmartphonesprovides an alternative to dedicated physical devices. To authenticate, people can use their personal access codes to the device (i.e. something that only the individual user knows) plus a one-time-valid, dynamic passcode, typically consisting of 4 to 6 digits. The passcode can be sent to their mobile device[3]bySMSor can be generated by a one-time passcode-generator app. In both cases, the advantage of using a mobile phone is that there is no need for an additional dedicated token, as users tend to carry theirmobile devicesaround at all times.
Notwithstanding the popularity of SMS verification, security advocates have publicly criticized SMS verification,[14]and in July 2016, a United StatesNISTdraft guideline proposed deprecating it as a form of authentication.[15]A year later NIST reinstated SMS verification as a valid authentication channel in the finalized guideline.[16]
As early as 2011, Duo Security was offeringpush notificationsfor MFA via a mobile app.[17]In 2016 and 2017 respectively, both Google and Apple started offering user two-step authentication with push notifications[4]as an alternative method.[18][19]
Security of mobile-delivered security tokens fully depends on the mobile operator's operational security and can be easily breached by wiretapping orSIM cloningby national security agencies.[20]
Advantages:
Disadvantages:
ThePayment Card Industry (PCI)Data Security Standard, requirement 8.3, requires the use of MFA for all remote network access that originates from outside the network to a Card Data Environment (CDE).[24]Beginning with PCI-DSS version 3.2, the use of MFA is required for all administrative access to the CDE, even if the user is within a trusted network.
The secondPayment Services Directiverequires "strong customer authentication" on most electronic payments in theEuropean Economic Areasince September 14, 2019.[25]
In India, theReserve Bank of Indiamandated two-factor authentication for all online transactions made using a debit or credit card using either a password or a one-time password sent overSMS. This requirement was removed in 2016 for transactions up to ₹2,000 after opting-in with the issuing bank.[26]Vendors such asUberhave been mandated by the bank to amend their payment processing systems in compliance with this two-factor authentication rollout.[27][28][29]
Details for authentication for federal employees and contractors in the U.S. are defined in Homeland Security Presidential Directive 12 (HSPD-12).[30]
IT regulatory standards for access to federal government systems require the use of multi-factor authentication to access sensitive IT resources, for example when logging on to network devices to perform administrative tasks[31]and when accessing any computer using a privileged login.[32]
NISTSpecial Publication 800-63-3 discusses various forms of two-factor authentication and provides guidance on using them in business processes requiring different levels of assurance.[33]
In 2005, the United States'Federal Financial Institutions Examination Councilissued guidance for financial institutions recommending financial institutions conduct risk-based assessments, evaluate customer awareness programs, and develop security measures to reliably authenticate customers remotely accessingonline financial services, officially recommending the use of authentication methods that depend on more than one factor (specifically, what a user knows, has, and is) to determine the user's identity.[34]In response to the publication, numerous authentication vendors began improperly promoting challenge-questions, secret images, and other knowledge-based methods as "multi-factor" authentication. Due to the resulting confusion and widespread adoption of such methods, on August 15, 2006, the FFIEC published supplemental guidelines—which state that by definition, a "true" multi-factor authentication system must use distinct instances of the three factors of authentication it had defined, and not just use multiple instances of a single factor.[35]
According to proponents, multi-factor authentication could drastically reduce the incidence of onlineidentity theftand other onlinefraud, because the victim's password would no longer be enough to give a thief permanent access to their information. However, many multi-factor authentication approaches remain vulnerable tophishing,[36]man-in-the-browser, andman-in-the-middle attacks.[37]Two-factor authentication in web applications are especially susceptible to phishing attacks, particularly in SMS and e-mails, and, as a response, many experts advise users not to share their verification codes with anyone,[38]and many web application providers will place an advisory in an e-mail or SMS containing a code.[39]
Multi-factor authentication may be ineffective[40]against modern threats, like ATM skimming, phishing, and malware.[41][vague][needs update?]
In May 2017,O2 Telefónica, a German mobile service provider, confirmed that cybercriminals had exploitedSS7vulnerabilities to bypass SMS based two-step authentication to do unauthorized withdrawals from users' bank accounts. The criminals firstinfectedthe account holder's computers in an attempt to steal their bank account credentials and phone numbers. Then the attackers purchased access to a fake telecom provider and set up a redirect for the victim's phone number to a handset controlled by them. Finally, the attackers logged into victims' online bank accounts and requested for the money on the accounts to be withdrawn to accounts owned by the criminals. SMS passcodes were routed to phone numbers controlled by the attackers and the criminals transferred the money out.[42]
An increasingly common approach to defeating MFA is to bombard the user with many requests to accept a log-in, until the user eventually succumbs to the volume of requests and accepts one.[43]This is called a multi-factor authentication fatigue attack (also MFA fatigue attack or MFA bombing) makes use ofsocial engineering.[44][45][46]When MFA applications are configured to send push notifications to end users, an attacker can send a flood of login attempts in the hope that a user will click on accept at least once.[44]
In 2022,Microsofthas deployed a mitigation against MFA fatigue attacks with their authenticator app.[47]
In September 2022Ubersecurity was breached by a member ofLapsus$using a multi-factor fatigue attack.[48][49]On March 24, 2023, YouTuberLinus Sebastiandeclared on theLinus Tech Tipschannel on theYouTubeplatform that he had suffered an Multi-factor authentication fatigue attack.[50]In early 2024, a small percentage ofAppleconsumers experienced a MFA fatigue attack that was caused by a hacker that bypassed the rate limit andCaptchaon Apple’s “Forgot Password” page.
Manymulti-factor authenticationproducts require users to deployclientsoftwareto make multi-factor authentication systems work. Some vendors have created separate installation packages fornetworklogin,Webaccesscredentials, andVPNconnectioncredentials. For such products, there may be four or five differentsoftwarepackages to push down to theclientPC in order to make use of thetokenorsmart card. This translates to four or five packages on which version control has to be performed, and four or five packages to check for conflicts with business applications. If access can be operated usingweb pages, it is possible to limit the overheads outlined above to a single application. With other multi-factor authentication technology such as hardware token products, no software must be installed by end-users.[citation needed]Some studies have shown that poorly implemented MFA recovery procedures can introduce new vulnerabilities that attackers may exploit.[51]
There are drawbacks to multi-factor authentication that are keeping many approaches from becoming widespread. Some users have difficulty keeping track of a hardware token or USB plug. Many users do not have the technical skills needed to install a client-side software certificate by themselves. Generally, multi-factor solutions require additional investment for implementation and costs for maintenance. Most hardware token-based systems are proprietary, and some vendors charge an annual fee per user. Deployment ofhardware tokensis logistically challenging. Hardwaretokensmay get damaged or lost, and issuance oftokensin large industries such as banking or even within large enterprises needs to be managed. In addition to deployment costs, multi-factor authentication often carries significant additional support costs.[citation needed]A 2008 survey[52]of over 120U.S. credit unionsby theCredit Union Journalreported on the support costs associated with two-factor authentication. In their report,software certificates and software toolbar approaches[clarification needed]were reported to have the highest support costs.
Research into deployments of multi-factor authentication schemes[53]has shown that one of the elements that tend to impact the adoption of such systems is the line of business of the organization that deploys the multi-factor authentication system. Examples cited include the U.S. government, which employs an elaborate system of physical tokens (which themselves are backed by robustPublic Key Infrastructure), as well as private banks, which tend to prefer multi-factor authentication schemes for their customers that involve more accessible, less expensive means of identity verification, such as an app installed onto a customer-owned smartphone. Despite the variations that exist among available systems that organizations may have to choose from, once a multi-factor authentication system is deployed within an organization, it tends to remain in place, as users invariably acclimate to the presence and use of the system and embrace it over time as a normalized element of their daily process of interaction with their relevant information system.
While the perception is that multi-factor authentication is within the realm of perfect security, Roger Grimes writes[54]that if not properly implemented and configured, multi-factor authentication can in fact be easily defeated.
In 2013,Kim Dotcomclaimed to have invented two-factor authentication in a 2000 patent,[55]and briefly threatened to sue all the major web services. However, the European Patent Office revoked his patent[56]in light of an earlier 1998 U.S. patent held by AT&T.[57]
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LinOTPis Linux-based software to manage authentication devices fortwo-factor authenticationwithone time passwords.
It is implemented as a web service based on the python frameworkPylons. Thus it requires a web server to
run in.
LinOTP is mainly developed by the German company KeyIdentity GmbH. Its core components are licensed under theAffero General Public License.
It is an open source authentication server certified[2]by theOATH initiative for open authenticationfor its 2.4 version.
As a web service, LinOTP provides aREST-like web API.[3]All functions can be accessed via Pylons controllers. Responses are returned as aJSONobject.
LinOTP is designed in a modular way, enabling user store modules and token modules. Thus, it is capable of supporting a wide range of different tokens.[4]
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The following is a generalcomparison of OTP applicationsthat are used to generateone-time passwordsfortwo-factor authentication(2FA) systems using thetime-based one-time password(TOTP) or theHMAC-based one-time password(HOTP) algorithms.
by 2Stable[45]
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HMAC-based one-time password(HOTP) is aone-time password(OTP) algorithm based onHMAC. It is a cornerstone of theInitiative for Open Authentication(OATH).
HOTP was published as an informationalIETFRFC4226in December 2005, documenting the algorithm along with a Java implementation. Since then, the algorithm has been adopted by many companies worldwide (see below). The HOTP algorithm is a freely availableopen standard.
The HOTP algorithm provides a method of authentication by symmetric generation of human-readable passwords, orvalues, each used for only one authentication attempt. The one-time property leads directly from the single use of each counter value.
Parties intending to use HOTP must establish someparameters; typically these are specified by the authenticator, and either accepted or not by the authenticated entity:
Both parties compute the HOTP value derived from the secret keyKand the counterC. Then the authenticator checks its locally generated value against the value supplied by the authenticated.
The authenticator and the authenticated entity increment the counterCindependently. Since the authenticated entity may increment the counter more than the authenticator,RFC4226recommends a resynchronization protocol. It proposes that the authenticator repeatedly try verification ahead of their counter through a window of sizes. The authenticator's counter continues forward of the value at which verification succeeds, and requires no actions by the authenticated entity.
To protect against brute-force attacks targeting the small size of HOTP values, the RFC also recommends implementing persistent throttling of HOTP verification. This can be achieved by either locking out verification after a small number of failed attempts, or by linearly increasing the delay after each failed attempt.
6-digit codes are commonly provided by proprietary hardware tokens from a number of vendors informing the default value ofd. Truncation extracts 31bitsorlog10(231)≈9.3{\textstyle \log _{10}(2^{31})\approx 9.3}decimal digits, meaning thatdcan be at most 10, with the 10th digit adding less variation, taking values of 0, 1, and 2 (i.e., 0.3 digits).
After verification, the authenticator can authenticate itself simply by generating the next HOTP value, returning it, and then the authenticated can generate their own HOTP value to verify it. Note that counters are guaranteed to be synchronised at this point in the process.
TheHOTP valueis the human-readable design output, ad-digit decimal number (without omission of leading 0s):
That is, the value is thedleast significant base-10 digits of HOTP.
HOTPis a truncation of theHMACof the counterC(under the keyKand hash functionH):
where the counterCmust be usedbig-endian.
Truncation first takes the 4 least significant bits of theMACand uses them as a byte offseti:
where ":" is used to extract bits from a starting bit number up to and including an ending bit number, where these bit numbers are 0-origin. The use of "19" in the above formula relates to the size of the output from the hash function. With the default of SHA-1, the output is20bytes, and so the last byte is byte 19 (0-origin).
That indexiis used to select 31 bits fromMAC, starting at biti× 8 + 1:
31 bits are a single bit short of a 4-byte word. Thus the value can be placed inside such a word without using the sign bit (the most significant bit). This is done to definitely avoid doing modular arithmetic on negative numbers, as this has many differing definitions and implementations.[1]
Both hardware and software tokens are available from various vendors, for some of them see references below.
Software tokens are available for (nearly) all major mobile/smartphoneplatforms (J2ME,[2]Android,[3]iPhone,[4]BlackBerry,[5]Maemo,[6]macOS,[7]and Windows Mobile[5]).
Although the early reception from some of the computer press was negative during 2004 and 2005,[8][9][10]after IETF adopted HOTP asRFC4226in December 2005, various vendors started to produce HOTP-compatible tokens and/or whole authentication solutions.
According to the article "Road Map: Replacing Passwords with OTP Authentication"[11]on strong authentication, published by Burton Group (a division ofGartner, Inc.) in 2010, "Gartner's expectation is that the hardwareOTPform factor will continue to enjoy modest growth whilesmartphoneOTPs will grow and become the default hardware platform over time."
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TheFIDO("Fast IDentity Online")Allianceis an open industry association launched in February 2013 whose stated mission is to develop and promoteauthenticationstandards that "help reduce the world’s over-reliance onpasswords".[1]FIDO addresses the lack of interoperability among devices that use strong authentication and reduces the problems users face creating and remembering multiple usernames and passwords.
FIDO supports a full range of authentication technologies, includingbiometricssuch asfingerprintandiris scanners,voiceandfacial recognition, as well as existing solutions and communications standards, such asTrusted Platform Modules(TPM),USBsecurity tokens, embedded Secure Elements (eSE),smart cards, andnear-field communication(NFC).[2]The USB security token device may be used to authenticate using a simple password (e.g. four-digitPIN) or by pressing a button. The specifications emphasize a device-centric model. Authentication over aninsecure channelhappens usingpublic-key cryptography. The user's device registers the user to a server by registering a public key. To authenticate the user, the device signs a challenge from the server using the private key that it holds. The keys on the device are unlocked by a local user gesture such as a biometric or pressing a button.
FIDO provides two types of user experiences depending on which protocol is used.[2]Both protocols define a common interface at the client for whatever local authentication method the user exercises.
The following open specifications may be obtained from the FIDO web site.[3]
The U2F 1.0 Proposed Standard (October 9, 2014) was the starting point for the specification known as FIDO 2.0 Proposed Standard (September 4, 2015). The latter was formally submitted to theWorld Wide Web Consortium(W3C) on November 12, 2015.[5]Subsequently, the first Working Draft of the W3C Web Authentication (WebAuthn) standard was published on May 31, 2016. The WebAuthn standard has been revised numerous times since then, becoming a W3C Recommendation on March 4, 2019.
Meanwhile the U2F 1.2 Proposed Standard (July 11, 2017) became the starting point for the Client to Authenticator Protocol 2.0 Proposed Standard, which was published on September 27, 2017. FIDO CTAP 2.0 complements W3C WebAuthn, both of which are in scope for theFIDO2 Project.
The FIDO2 Project is a joint effort between the FIDO Alliance and theWorld Wide Web Consortium(W3C) whose goal is to create strong authentication for the web. At its core, FIDO2 consists of the W3C Web Authentication (WebAuthn) standard and the FIDOClient to Authenticator Protocol2 (CTAP2).[6]FIDO2 is based upon previous work done by the FIDO Alliance, in particular theUniversal 2nd Factor(U2F) authentication standard.
Taken together, WebAuthn and CTAP specify a standardauthentication protocol[7]where the protocol endpoints consist of a user-controlledcryptographicauthenticator(such as a smartphone or a hardwaresecurity key) and a WebAuthn Relying Party (also called a FIDO2 server). A webuser agent(i.e., a web browser) together with a WebAuthn client form an intermediary between the authenticator and the relying party. A single WebAuthn client Device may support multiple WebAuthn clients. For example, a laptop may support multiple clients, one for each conforming user agent running on the laptop. A conforming user agent implements the WebAuthn JavaScript API.
As its name implies, theClient to Authenticator Protocol(CTAP) enables a conforming cryptographic authenticator to interoperate with a WebAuthn client. The CTAP specification refers to two protocol versions called CTAP1/U2F and CTAP2.[8]An authenticator that implements one of these protocols is typically referred to as a U2F authenticator or a FIDO2 authenticator, respectively. A FIDO2 authenticator that also implements the CTAP1/U2F protocol is backward compatible with U2F.
The invention of using a smartphone as a cryptographic authenticator on a computer network is claimed in US Patent 7,366,913 filed in 2002.[9]
[15]
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The list below includes the names of notable ofpassword managerswith their Wikipedia articles.
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Apassword manageris a software program to preventpassword fatiguebyautomatically generating,autofillingand storingpasswords.[1][2]It can do this forlocal applicationsorweb applicationssuch asonline shopsorsocial media.[3]Web browserstend to have a built-in password manager. Password managers typically require a user to create and remember a single password to unlock to access the stored passwords. Password managers can integratemulti-factor authentication.
The first password manager software designed to securely store passwords wasPassword Safecreated byBruce Schneier, which was released as a free utility on September 5, 1997.[4]Designed forMicrosoftWindows 95, Password Safe used Schneier'sBlowfishalgorithmto encrypt passwords and other sensitive data. Although Password Safe was released as a free utility, due toexport restrictions on cryptography from the United States, only U.S. and Canadian citizens and permanent residents were initially allowed to download it.[4]
As of October 2024[update], the built-in Google Password Manager inGoogle Chromebecame the most used password manager.[5]
Some applications store passwords as an unencrypted file, leaving the passwords easily accessible tomalwareor people attempted to steal personal information.
Some password managers require a user-selected master password orpassphraseto form thekeyused to encrypt passwords stored for the application to read. The security of this approach depends on the strength of the chosen password (which may be guessed through malware), and also that the passphrase itself is never stored locally where a malicious program or individual could read it. A compromised master password may render all of the protected passwords vulnerable, meaning that a single point of entry can compromise the confidentiality of sensitive information. This is known as asingle point of failure.
While password managers offer robust security for credentials, their effectiveness hinges on the user's device security. If a device is compromised by malware like Raccoon, which excels at stealing data, the password manager's protections can be nullified. Malware like keyloggers can steal the master password used to access the password manager, granting full access to all stored credentials. Clipboard sniffers can capture sensitive information copied from the manager, and some malware might even steal the encrypted password vault file itself. In essence, a compromised device with password-stealing malware can bypass the security measures of the password manager, leaving the stored credentials vulnerable.[6]
As with password authentication techniques,key loggingor acoustic cryptanalysis may be used to guess or copy the "master password". Some password managers attempt to usevirtual keyboardsto reduce this risk - though this is still vulnerable to key loggers.[7]that take the keystrokes and send what key was pressed to the person/people trying to access confidential information.
Cloud-based password managers offer a centralized location for storing login credentials. However, this approach raises security concerns. One potential vulnerability is a data breach at the password manager itself. If such an event were to occur, attackers could potentially gain access to a large number of user credentials.A 2022 security incident involving LastPassexemplifies this risk.[6]
Some password managers may include a password generator. Generated passwords may be guessable if the password manager uses a weak method ofrandomly generating a "seed"for all passwords generated by this program. There are documented cases, like the one withKasperskyPassword Manager in 2021, where a flaw in the password generation method resulted in predictable passwords.[8][9]
A 2014 paper by researchers atCarnegie Mellon Universityfound that while browsers refuse to autofill passwords if the login page protocol differs from when the password was saved (HTTPvs.HTTPS), some password managers insecurely filled passwords for the unencrypted (HTTP) version of saved passwords for encrypted (HTTPS) sites. Additionally, most managers lacked protection againstiframeandredirection-basedattacks, potentially exposing additional passwords whenpassword synchronizationwas used across multiple devices.[10]
Various high-profile websites have attempted to block password managers, often backing down when publicly challenged.[11][12][13]Reasons cited have included protecting againstautomated attacks, protecting againstphishing, blockingmalware, or simply denying compatibility. TheTrusteerclient security software fromIBMfeatures explicit options to block password managers.[14][15]
Such blocking has been criticized byinformation securityprofessionals as making users less secure.[13][15]The typical blocking implementation involves settingautocomplete='off'on the relevant passwordweb form.
This option is now consequently ignored onencrypted sites,[10]such asFirefox38,[16]Chrome34,[17]andSafarifrom about 7.0.2.[18]
In recent years, some websites have made it harder for users to rely on password managers by disabling features like password autofill or blocking the ability to paste into password fields. Companies like T-Mobile, Barclaycard, and Western Union have implemented these restrictions, often citing security concerns such as malware prevention, phishing protection, or reducing automated attacks. However, cybersecurity experts have criticized these measures, arguing they can backfire by encouraging users to reuse weak passwords or rely on memory alone—ultimately making accounts more vulnerable. Some organizations, such asBritish Gas, have reversed these restrictions after public feedback, but the practice still persists on many websites.[19]
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Incommunicationsandinformation processing,codeis a system of rules to convertinformation—such as aletter,word, sound, image, orgesture—into another form, sometimesshortenedorsecret, for communication through acommunication channelor storage in astorage medium. An early example is an invention oflanguage, which enabled a person, throughspeech, to communicate what they thought, saw, heard, or felt to others. But speech limits the range of communication to the distance a voice can carry and limits the audience to those present when the speech is uttered. The invention ofwriting, which converted spoken language intovisualsymbols, extended the range of communication across space andtime.
The process ofencodingconverts information from asourceinto symbols for communication or storage.Decodingis the reverse process, converting code symbols back into a form that the recipient understands, such as English, Spanish, etc.
One reason for coding is to enable communication in places where ordinaryplain language, spoken or written, is difficult or impossible. For example,semaphore, where the configuration offlagsheld by a signaler or the arms of asemaphore towerencodes parts of the message, typically individual letters, and numbers. Another person standing a great distance away can interpret the flags and reproduce the words sent.
Ininformation theoryandcomputer science, a code is usually considered as analgorithmthat uniquely representssymbolsfrom some sourcealphabet, byencodedstrings, which may be in some other target alphabet. An extension of the code for representing sequences of symbols over the source alphabet is obtained by concatenating the encoded strings.
Before giving a mathematically precise definition, this is a brief example. The mapping
is a code, whose source alphabet is the set{a,b,c}{\displaystyle \{a,b,c\}}and whose target alphabet is the set{0,1}{\displaystyle \{0,1\}}. Using the extension of the code, the encoded string 0011001 can be grouped into codewords as 0 011 0 01, and these in turn can be decoded to the sequence of source symbolsacab.
Using terms fromformal language theory, the precise mathematical definition of this concept is as follows: let S and T be two finite sets, called the source and targetalphabets, respectively. AcodeC:S→T∗{\displaystyle C:\,S\to T^{*}}is atotal functionmapping each symbol from S to asequence of symbolsover T. TheextensionC′{\displaystyle C'}ofC{\displaystyle C}, is ahomomorphismofS∗{\displaystyle S^{*}}intoT∗{\displaystyle T^{*}}, which naturally maps each sequence of source symbols to a sequence of target symbols.
In this section, we consider codes that encode each source (clear text) character by acode wordfrom some dictionary, andconcatenationof such code words give us an encoded string. Variable-length codes are especially useful when clear text characters have different probabilities; see alsoentropy encoding.
Aprefix codeis a code with the "prefix property": there is no valid code word in the system that is aprefix(start) of any other valid code word in the set.Huffman codingis the most known algorithm for deriving prefix codes. Prefix codes are widely referred to as "Huffman codes" even when the code was not produced by a Huffman algorithm. Other examples of prefix codes aretelephone country codes, the country and publisher parts ofISBNs, and the Secondary Synchronization Codes used in theUMTSWCDMA3G Wireless Standard.
Kraft's inequalitycharacterizes the sets of codeword lengths that are possible in a prefix code. Virtually any uniquely decodable one-to-many code, not necessarily a prefix one, must satisfy Kraft's inequality.
Codes may also be used to represent data in a way more resistant to errors in transmission or storage. This so-callederror-correcting codeworks by including carefully crafted redundancy with the stored (or transmitted) data. Examples includeHamming codes,Reed–Solomon,Reed–Muller,Walsh–Hadamard,Bose–Chaudhuri–Hochquenghem,Turbo,Golay,algebraic geometry codes,low-density parity-check codes, andspace–time codes.
Error detecting codes can be optimised to detectburst errors, orrandom errors.
A cable code replaces words (e.g.shiporinvoice) with shorter words, allowing the same information to be sent with fewercharacters, more quickly, and less expensively.
Codes can be used for brevity. Whentelegraphmessages were the state of the art in rapid long-distance communication, elaborate systems ofcommercial codesthat encoded complete phrases into single mouths (commonly five-minute groups) were developed, so that telegraphers became conversant with such "words" asBYOXO("Are you trying to weasel out of our deal?"),LIOUY("Why do you not answer my question?"),BMULD("You're a skunk!"), orAYYLU("Not clearly coded, repeat more clearly.").Code wordswere chosen for various reasons:length,pronounceability, etc. Meanings were chosen to fit perceived needs: commercial negotiations, military terms for military codes, diplomatic terms for diplomatic codes, any and all of the preceding for espionage codes. Codebooks and codebook publishers proliferated, including one run as a front for the AmericanBlack Chamberrun byHerbert Yardleybetween the First and Second World Wars. The purpose of most of these codes was to save on cable costs. The use of data coding fordata compressionpredates the computer era; an early example is the telegraphMorse codewhere more-frequently used characters have shorter representations. Techniques such asHuffman codingare now used by computer-basedalgorithmsto compress large data files into a more compact form for storage or transmission.
Character encodings are representations of textual data. A given character encoding may be associated with a specific character set (the collection of characters which it can represent), though some character sets have multiple character encodings and vice versa. Character encodings may be broadly grouped according to the number of bytes required to represent a single character: there are single-byte encodings,multibyte(also called wide) encodings, andvariable-width(also called variable-length) encodings. The earliest character encodings were single-byte, the best-known example of which isASCII. ASCII remains in use today, for example inHTTP headers. However, single-byte encodings cannot model character sets with more than 256 characters. Scripts that require large character sets such asChinese, Japanese and Koreanmust be represented with multibyte encodings. Early multibyte encodings were fixed-length, meaning that although each character was represented by more than one byte, all characters used the same number of bytes ("word length"), making them suitable for decoding with a lookup table. The final group, variable-width encodings, is a subset of multibyte encodings. These use more complex encoding and decoding logic to efficiently represent large character sets while keeping the representations of more commonly used characters shorter or maintaining backward compatibility properties. This group includesUTF-8, an encoding of theUnicodecharacter set; UTF-8 is the most common encoding of text media on the Internet.
Biologicalorganisms contain genetic material that is used to control their function and development. This isDNA, which contains units namedgenesfrom whichmessenger RNAis derived. This in turn producesproteinsthrough agenetic codein which a series of triplets (codons) of four possiblenucleotidescan be translated into one of twenty possibleamino acids. A sequence of codons results in a corresponding sequence of amino acids that form a protein molecule; a type of codon called astop codonsignals the end of the sequence.
Inmathematics, aGödel codeis the basis for the proof ofGödel'sincompleteness theorem. Here, the idea is to mapmathematical notationto anatural number(using aGödel numbering).
There are codes using colors, liketraffic lights, thecolor codeemployed to mark the nominal value of theelectrical resistorsor that of the trashcans devoted to specific types of garbage (paper, glass, organic, etc.).
Inmarketing,couponcodes can be used for a financial discount or rebate when purchasing a product from a (usual internet) retailer.
In military environments, specific sounds with thecornetare used for different uses: to mark some moments of the day, to command the infantry on the battlefield, etc.
Communication systems for sensory impairments, such assign languagefor deaf people andbraillefor blind people, are based on movement or tactile codes.
Musical scoresare the most common way to encodemusic.
Specific games have their own code systems to record the matches, e.g.chess notation.
In thehistory of cryptography,codeswere once common for ensuring the confidentiality of communications, althoughciphersare now used instead.
Secret codes intended to obscure the real messages, ranging from serious (mainlyespionagein military, diplomacy, business, etc.) to trivial (romance, games) can be any kind of imaginative encoding:flowers, game cards, clothes, fans, hats, melodies, birds, etc., in which the sole requirement is the pre-agreement on the meaning by both the sender and the receiver.
Other examples of encoding include:
Other examples of decoding include:
Acronymsand abbreviations can be considered codes, and in a sense, alllanguagesandwriting systemsare codes for human thought.
International Air Transport Association airport codesare three-letter codes used to designate airports and used forbag tags.Station codesare similarly used on railways but are usually national, so the same code can be used for different stations if they are in different countries.
Occasionally, a code word achieves an independent existence (and meaning) while the original equivalent phrase is forgotten or at least no longer has the precise meaning attributed to the code word. For example, '30' was widely used injournalismto mean "end of story", and has been used inother contextsto signify "the end".[1][2]
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Communicationis commonly defined as the transmission ofinformation. Its precise definition is disputed and there are disagreements about whetherunintentionalor failed transmissions are included and whether communication not only transmitsmeaningbut also creates it.Models of communicationare simplified overviews of its main components and their interactions. Many models include the idea that a source uses acodingsystem to express information in the form of a message. The message is sent through achannelto a receiver who has to decode it to understand it. The main field of inquiry investigating communication is calledcommunication studies.
A common way to classify communication is by whether information is exchanged between humans, members of other species, or non-living entities such as computers. For human communication, a central contrast is betweenverbalandnon-verbal communication. Verbal communication involves the exchange of messages inlinguisticform, including spoken and written messages as well assign language. Non-verbal communication happens without the use of alinguistic system, for example, usingbody language,touch, and facial expressions. Another distinction is betweeninterpersonal communication, which happens between distinct persons, andintrapersonal communication, which is communication with oneself.Communicative competenceis the ability to communicate well and applies to the skills of formulating messages and understanding them.
Non-human forms of communication includeanimalandplant communication. Researchers in this field often refine their definition of communicative behavior by including the criteria that observable responses are present and that the participants benefit from the exchange. Animal communication is used in areas likecourtshipand mating, parent–offspring relations, navigation, and self-defense. Communication through chemicals is particularly important for the relatively immobile plants. For example,mapletrees release so-calledvolatile organic compoundsinto the air to warn other plants of aherbivoreattack. Most communication takes place between members of the same species. The reason is that its purpose is usually some form of cooperation, which is not as common between different species.Interspecies communicationhappens mainly in cases ofsymbioticrelationships. For instance, many flowers use symmetrical shapes and distinctive colors to signal to insects wherenectaris located. Humans engage in interspecies communication when interacting withpetsandworking animals.
Human communication has a longhistoryand how people exchange information has changed over time. These changes were usually triggered by the development of newcommunication technologies. Examples are the invention ofwriting systems, the development of mass printing, the use of radio and television, and the invention of the internet. The technological advances also led to new forms of communication, such as theexchange of data between computers.
The wordcommunicationhas its root in theLatinverbcommunicare, which means'to share'or'to make common'.[1]Communication is usually understood as the transmission of information:[2]amessageis conveyed from a sender to a receiver using some medium, such as sound, written signs, bodily movements, or electricity.[3]Sender and receiver are often distinct individuals but it is also possible for an individual to communicate with themselves. In some cases, sender and receiver are not individuals but groups like organizations, social classes, or nations.[4]In a different sense, the termcommunicationrefers to the message that is being communicated or to thefield of inquiry studying communicational phenomena.[5]
The precise characterization of communication is disputed. Many scholars have raised doubts that any single definition can capture the term accurately. These difficulties come from the fact that the term is applied to diverse phenomena in different contexts, often with slightly different meanings.[6]The issue of the right definition affects the research process on many levels. This includes issues like whichempirical phenomenaare observed, how they are categorized, whichhypothesesand laws are formulated as well as how systematic theories based on these steps are articulated.[7]
Some definitions are broad and encompass unconscious and non-humanbehavior.[8]Under a broad definition, many animals communicate within their own species and flowers communicate by signaling the location of nectar to bees through their colors and shapes.[9]Other definitions restrict communication toconsciousinteractions among human beings.[10]Some approaches focus on the use of symbols and signs while others stress the role of understanding, interaction, power, or transmission of ideas. Various characterizations see the communicator'sintentto send a message as a central component. In this view, the transmission of information is not sufficient for communication if it happens unintentionally.[11]A version of this view is given by philosopherPaul Grice, who identifies communication withactionsthat aim to make the recipient aware of the communicator's intention.[12]One question in this regard is whether only successful transmissions of information should be regarded as communication.[13]For example, distortion may interfere with and change the actual message from what was originally intended.[14]A closely related problem is whether acts of deliberatedeceptionconstitute communication.[15]
According to a broad definition by literary criticI. A. Richards, communication happens when onemindacts upon its environment to transmit its ownexperienceto another mind.[16]Another interpretation is given by communication theoristsClaude ShannonandWarren Weaver, who characterize communication as a transmission of information brought about by the interaction of several components, such as a source, a message, an encoder, a channel, a decoder, and a receiver.[17]The transmission view is rejected by transactional and constitutive views, which hold that communication is not just about the transmission of information but also about the creation of meaning. Transactional and constitutive perspectives hold that communication shapes the participant's experience by conceptualizing the world and making sense of their environment and themselves.[18]Researchers studying animal and plant communication focus less on meaning-making. Instead, they often define communicative behavior as having other features, such as playing a beneficial role in survival and reproduction, or having an observable response.[19]
Models of communication areconceptualrepresentations of the process of communication.[20]Their goal is to provide a simplified overview of its main components. This makes it easier for researchers to formulate hypotheses, apply communication-related concepts to real-world cases, and testpredictions.[21]Due to their simplified presentation, they may lack the conceptual complexity needed for a comprehensive understanding of all the essential aspects of communication. They are usually presented visually in the form ofdiagramsshowing the basic components and their interaction.[22]
Models of communication are often categorized based on their intended applications and how they conceptualize communication. Some models are general in the sense that they are intended for all forms of communication. Specialized models aim to describe specific forms, such as models ofmass communication.[23]
One influential way to classify communication is to distinguish between linear transmission, interaction, and transaction models.[24]Linear transmission models focus on how a sender transmitsinformationto a receiver. They arelinearbecause this flow of information only goes in a single direction.[25]This view is rejected by interaction models, which include afeedbackloop. Feedback is needed to describe many forms of communication, such as a conversation, where the listener may respond to a speaker by expressing their opinion or by asking for clarification. Interaction models represent the process as a form oftwo-way communicationin which the communicators take turns sending and receiving messages.[26]Transaction models further refine this picture by allowing representations of sending and responding at the same time. This modification is needed to describe how the listener can give feedback in a face-to-face conversation while the other person is talking. Examples arenon-verbal feedbackthroughbody postureandfacial expression. Transaction models also hold that meaning is produced during communication and does not exist independently of it.[27]
All the early models, developed in the middle of the 20th century, are linear transmission models.Lasswell's model, for example, is based on five fundamental questions: "Who?", "Says what?", "In which channel?", "To whom?", and "With what effect?".[28]The goal of these questions is to identify the basic components involved in the communicative process: the sender, the message, thechannel, thereceiver, and the effect.[29]Lasswell's model was initially only conceived as a model of mass communication, but it has been applied to other fields as well. Some communication theorists, like Richard Braddock, have expanded it by including additional questions, like "Under what circumstances?" and "For what purpose?".[30]
TheShannon–Weaver modelis another influential linear transmission model.[31]It is based on the idea that a source creates a message, which is then translated into asignalby a transmitter.Noisemay interfere with and distort the signal. Once the signal reaches the receiver, it is translated back into a message and made available to the destination. For a landline telephone call, the person calling is the source and their telephone is the transmitter. The transmitter translates the message into an electrical signal that travels through the wire, which acts as the channel. The person taking the call is the destination and their telephone is the receiver.[32]The Shannon–Weaver model includes an in-depth discussion of how noise can distort the signal and how successful communication can be achieved despite noise. This can happen by making the message partiallyredundantso that decoding is possible nonetheless.[33]Other influential linear transmission models includeGerbner's modelandBerlo's model.[34]
The earliest interaction model was developed by communication theoristWilbur Schramm.[35]He states that communication starts when a source has an idea and expresses it in the form of a message. This process is calledencodingand happens using acode, i.e. asign systemthat is able to express the idea, for instance, throughvisualor auditory signs.[36]The message is sent to a destination, who has to decode and interpret it to understand it.[37]In response, they formulate their own idea, encode it into a message, and send it back as a form of feedback. Another innovation ofSchramm's modelis that previous experience is necessary to be able to encode and decode messages. For communication to be successful, the fields of experience of source and destination have to overlap.[38]
The first transactional model was proposed by communication theoristDean Barnlundin 1970.[39]He understands communication as "the production of meaning, rather than the production of messages".[40]Its goal is to decrease uncertainty and arrive at a sharedunderstanding.[41]This happens in response to external and internal cues. Decoding is the process of ascribing meaning to them and encoding consists in producing newbehavioralcues as a response.[42]
There are many forms ofhuman communication. A central distinction is whether language is used, as in the contrast between verbal and non-verbal communication. A further distinction concerns whether one communicates with others or with oneself, as in the contrast betweeninterpersonalandintrapersonal communication.[43]Forms of human communication are also categorized by their channel or the medium used to transmit messages.[44]The field studying human communication is known as anthroposemiotics.[45]
Verbal communication is the exchange of messages inlinguisticform, i.e., by means oflanguage.[46]In colloquial usage, verbal communication is sometimes restricted tooral communicationand may exclude writing and sign language. However, in academic discourse, the term is usually used in a wider sense, encompassing any form of linguistic communication, whether through speech, writing, or gestures.[47]Some of the challenges in distinguishing verbal from non-verbal communication come from the difficulties in defining what exactlylanguagemeans. Language is usually understood as a conventional system ofsymbolsand rules used for communication. Such systems are based on a set of simple units of meaning that can be combined to express more complex ideas. The rules for combining the units into compound expressions are calledgrammar.Wordsare combined to formsentences.[48]
One hallmark of human language, in contrast to animal communication, lies in its complexity and expressive power. Human language can be used to refer not just toconcrete objectsin the here-and-now but also to spatially and temporally distant objects and toabstract ideas.[49]Humans have a natural tendency toacquire their native language in childhood. They are also able to learn other languages later in life assecond languages. However, this process is less intuitive and often does not result in the same level oflinguistic competence.[50]The academic discipline studying language is calledlinguistics. Its subfields includesemantics(the study of meaning),morphology(the study of word formation),syntax(the study of sentence structure),pragmatics(the study of language use), andphonetics(the study of basic sounds).[51]
A central contrast among languages is betweennaturaland artificial orconstructed languages. Natural languages, likeEnglish,Spanish, andJapanese, developed naturally and for the most part unplanned in the course of history. Artificial languages, likeEsperanto,Quenya,C++, and the language offirst-order logic, are purposefully designed from the ground up.[52]Most everyday verbal communication happens using natural languages. Central forms of verbal communication are speech and writing together with their counterparts of listening and reading.[53]Spoken languages use sounds to producesignsand transmit meaning while for writing, the signs are physically inscribed on a surface.[54]Sign languages, likeAmerican Sign LanguageandNicaraguan Sign Language, are another form of verbal communication. They rely on visual means, mostly by using gestures with hands and arms, to form sentences and convey meaning.[55]
Verbal communication serves various functions. One key function is to exchange information, i.e. an attempt by the speaker to make the audience aware of something, usually of an external event. But language can also be used to express the speaker's feelings and attitudes. A closely related role is to establish and maintain social relations with other people. Verbal communication is also utilized to coordinate one's behavior with others and influence them. In some cases, language is not employed for an external purpose but only forentertainmentor personal enjoyment.[56]Verbal communication further helps individualsconceptualizethe world around them and themselves. This affects how perceptions of external events are interpreted, how things are categorized, and how ideas are organized and related to each other.[57]
Non-verbal communication is the exchange of information through non-linguistic modes, like facial expressions,gestures, andpostures.[58]However, not every form of non-verbal behavior constitutes non-verbal communication. Some theorists, likeJudee Burgoon, hold that it depends on the existence of a socially shared coding system that is used to interpret the meaning of non-verbal behavior.[59]Non-verbal communication has many functions. It frequently contains information about emotions, attitudes, personality, interpersonal relations, and private thoughts.[60]
Non-verbal communication often happens unintentionally and unconsciously, likesweatingorblushing, but there are also conscious intentional forms, like shaking hands orraising a thumb.[61]It often happens simultaneously with verbal communication and helps optimize the exchange through emphasis and illustration or by adding additional information. Non-verbal cues can clarify the intent behind a verbal message.[62]Using multiplemodalitiesof communication in this way usually makes communication more effective if the messages of each modality are consistent.[63]However, in some cases different modalities can contain conflicting messages. For example, a person may verbally agree with a statement but press their lips together, thereby indicating disagreement non-verbally.[64]
There are many forms of non-verbal communication. They includekinesics,proxemics,haptics,paralanguage,chronemics, and physical appearance.[65]Kinesics studies the role of bodily behavior in conveying information. It is commonly referred to asbody language, even though it is, strictly speaking, not a language but rather non-verbal communication. It includes many forms, like gestures, postures, walking styles, and dance.[66]Facial expressions, like laughing, smiling, and frowning, all belong to kinesics and are expressive and flexible forms of communication.[67]Oculesics is another subcategory of kinesics in regard to the eyes. It covers questions like how eye contact, gaze, blink rate, and pupil dilation form part of communication.[68]Some kinesic patterns are inborn and involuntary, like blinking, while others are learned and voluntary, like giving amilitary salute.[69]
Proxemics studies how personal space is used in communication. The distance between the speakers reflects their degree of familiarity and intimacy with each other as well as their social status.[70]Haptics examines how information is conveyed using touching behavior, like handshakes, holding hands, kissing, or slapping. Meanings linked to haptics include care, concern, anger, and violence. For instance, handshaking is often seen as a symbol of equality and fairness, while refusing to shake hands can indicate aggressiveness. Kissing is another form often used to show affection and erotic closeness.[71]
Paralanguage, also known as vocalics, encompasses non-verbal elements in speech that convey information. Paralanguage is often used to express the feelings and emotions that the speaker has but does not explicitly stated in the verbal part of the message. It is not concerned with the words used but with how they are expressed. This includes elements like articulation, lip control, rhythm, intensity, pitch, fluency, and loudness.[72]For example, saying something loudly and in a high pitch conveys a different meaning on the non-verbal level than whispering the same words. Paralanguage is mainly concerned with spoken language but also includes aspects of written language, like the use of colors and fonts as well as spatial arrangement in paragraphs and tables.[73]Non-linguistic sounds may also convey information;cryingindicates that an infant is distressed, andbabblingconveys information about infant health and well-being.[74]
Chronemics concerns the use of time, such as what messages are sent by being on time versus late for a meeting.[75]The physical appearance of the communicator, such as height, weight, hair, skin color, gender, clothing, tattooing, and piercing, also carries information.[76]Appearance is an important factor for first impressions but is more limited as a mode of communication since it is less changeable.[77]Some forms of non-verbal communication happen using such artifacts as drums, smoke, batons, traffic lights, and flags.[78]
Non-verbal communication can also happen through visualmedialikepaintingsanddrawings. They can express what a person or an object looks like and can also convey other ideas and emotions. In some cases, this type of non-verbal communication is used in combination with verbal communication, for example, whendiagramsormapsemploy labels to include additional linguistic information.[79]
Traditionally, most research focused on verbal communication. However, this paradigm began to shift in the 1950s when research interest in non-verbal communication increased and emphasized its influence.[80]For example, many judgments about the nature and behavior of other people are based on non-verbal cues.[81]It is further present in almost every communicative act to some extent and certain parts of it are universally understood.[82]These considerations have prompted some communication theorists, likeRay Birdwhistell, to claim that the majority of ideas and information is conveyed this way.[83]It has also been suggested that human communication is at its core non-verbal and that words can only acquire meaning because of non-verbal communication.[84]The earliest forms of human communication, such as crying and babbling, are non-verbal.[85]Some basic forms of communication happen even before birth between mother and embryo and include information about nutrition and emotions.[86]Non-verbal communication is studied in various fields besides communication studies, like linguistics,semiotics,anthropology, andsocial psychology.[87]
Interpersonal communication is communication between distinct people. Its typical form isdyadic communication, i.e. between two people, but it can also refer tocommunication within groups.[88]It can be planned or unplanned and occurs in many forms, like when greeting someone, during salary negotiations, or when making a phone call.[89]Some communication theorists, like Virginia M. McDermott, understand interpersonal communication as afuzzy conceptthat manifests in degrees.[90]In this view, an exchange varies in how interpersonal it is based on several factors. It depends on how many people are present, and whether it happens face-to-face rather than through telephone or email. A further factor concerns the relation between the communicators:[91]group communication and mass communication are less typical forms of interpersonal communication and some theorists treat them as distinct types.[92]
Interpersonal communication can be synchronous or asynchronous. For asynchronous communication, the parties take turns in sending and receiving messages. This occurs when exchanging letters or emails. For synchronous communication, both parties send messages at the same time.[93]This happens when one person is talking while the other person sends non-verbal messages in response signaling whether they agree with what is being said.[94]Some communication theorists, like Sarah Trenholm and Arthur Jensen, distinguish between content messages and relational messages. Content messages express the speaker's feelings toward the topic of discussion. Relational messages, on the other hand, demonstrate the speaker's feelings toward their relation with the other participants.[95]
Various theories of the function of interpersonal communication have been proposed. Some focus on how it helps people make sense of their world and create society. Others hold that its primary purpose is to understand why other people act the way they do and to adjust one's behavior accordingly.[96]A closely related approach is to focus on information and see interpersonal communication as an attempt to reduce uncertainty about others and external events.[97]Other explanations understand it in terms of theneedsit satisfies. This includes the needs of belonging somewhere, being included, being liked, maintaining relationships, and influencing the behavior of others.[98]On a practical level, interpersonal communication is used to coordinate one's actions with the actions of others to get things done.[99]Research on interpersonal communication includes topics like how people build, maintain, and dissolve relationships through communication. Other questions are why people choose one message rather than another and what effects these messages have on the communicators and their relation. A further topic is how to predict whether two people would like each other.[100]
Intrapersonal communication is communication with oneself.[101]In some cases this manifests externally, such as when engaged in amonologue, taking notes, highlighting a passage, and writing a diary or a shopping list. But many forms of intrapersonal communication happen internally in the form of an inner exchange with oneself, as when thinking about something ordaydreaming.[102]Closely related to intrapersonal communication is communication that takes place within an organism below the personal level, such as exchange of information between organs or cells.[103]
Intrapersonal communication can be triggered by internal and external stimuli. It may happen in the form of articulating a phrase before expressing it externally. Other forms are to make plans for the future and to attempt to process emotions to calm oneself down in stressful situations.[104]It can help regulate one's own mental activity and outward behavior as well as internalize cultural norms and ways of thinking.[105]External forms of intrapersonal communication can aid one's memory. This happens, for example, when making a shopping list. Another use is to unravel difficult problems, as when solving a complex mathematical equation line by line. New knowledge can also be internalized this way, such as when repeating new vocabulary to oneself. Because of these functions, intrapersonal communication can be understood as "an exceptionally powerful and pervasive tool for thinking."[106]
Based on its role inself-regulation, some theorists have suggested that intrapersonal communication is more basic than interpersonal communication. Young children sometimes useegocentric speechwhile playing in an attempt to direct their own behavior. In this view, interpersonal communication only develops later when the child moves from their early egocentric perspective to a more social perspective.[107]A different explanation holds that interpersonal communication is more basic since it is first used by parents to regulate what their child does. Once the child has learned this, they can apply the same technique to themselves to get more control over their own behavior.[108]
For communication to be successful, the message has to travel from the sender to the receiver. Thechannelis the way this is accomplished. It is not concerned with the meaning of the message but only with the technical means of how the meaning is conveyed.[109]Channels are often understood in terms of thesensesused to perceive the message, i.e. hearing, seeing, smelling, touching, and tasting.[110]But in the widest sense, channels encompass any form of transmission, including technological means like books, cables, radio waves, telephones, or television.[111]Naturally transmitted messages usually fade rapidly whereas some messages using artificial channels have a much longer lifespan, as in the case of books or sculptures.[112]
The physical characteristics of a channel have an impact on the code and cues that can be used to express information. For example, typical telephone calls are restricted to the use of verbal language and paralanguage but exclude facial expressions. It is often possible to translate messages from one code into another to make them available to a different channel. An example is writing down a spoken message or expressing it using sign language.[113]
The transmission of information can occur through multiple channels at once. For example, face-to-face communication often combines the auditory channel to convey verbal information with the visual channel to transmit non-verbal information using gestures and facial expressions. Employing multiple channels can enhance the effectiveness of communication by helping the receiver better understand the subject matter.[114]The choice of channels often matters since the receiver's ability to understand may vary depending on the chosen channel. For instance, a teacher may decide to present some information orally and other information visually, depending on the content and the student's preferred learning style. This underlines the role of amedia-adequateapproach.[115]
Communicative competence is the ability to communicate effectively or to choose the appropriate communicative behavior in a given situation.[116]It concerns what to say, when to say it, and how to say it.[117]It further includes the ability to receive and understand messages.[118]Competenceis often contrasted withperformancesince competence can be present even if it is not exercised, while performance consists in the realization of this competence.[119]However, some theorists reject a stark contrast and hold that performance is the observable part and is used to infer competence in relation to future performances.[120]
Two central components of communicative competence areeffectivenessand appropriateness.[121]Effectiveness is the degree to which the speaker achieves their desired outcomes or the degree to which preferred alternatives are realized.[122]This means that whether a communicative behavior is effective does not just depend on the actual outcome but also on the speaker's intention, i.e. whether this outcome was what they intended to achieve. Because of this, some theorists additionally require that the speaker be able to give an explanation of why they engaged in one behavior rather than another.[123]Effectiveness is closely related toefficiency, the difference being that effectiveness is about achieving goals while efficiency is about using few resources (such as time, effort, and money) in the process.[124]
Appropriateness means that the communicative behavior meets social standards and expectations.[125]Communication theorist Brian H. Spitzberg defines it as "the perceived legitimacy or acceptability of behavior or enactments in a given context".[126]This means that the speaker is aware of the social and cultural context in order to adapt and express the message in a way that is considered acceptable in the given situation.[127]For example, to bid farewell to their teacher, a student may use the expression "Goodbye, sir" but not the expression "I gotta split, man", which they may use when talking to a peer.[128]To be both effective and appropriate means to achieve one's preferred outcomes in a way that follows social standards and expectations.[129]Some definitions of communicative competence put their main emphasis on either effectiveness or appropriateness while others combine both features.[130]
Many additional components of communicative competence have been suggested, such asempathy, control, flexibility, sensitivity, and knowledge.[131]It is often discussed in terms of the individual skills employed in the process, i.e. the specific behavioral components that make up communicative competence.[132]Message production skills include reading and writing. They are correlated with the reception skills of listening and reading.[133]There are both verbal and non-verbal communication skills.[134]For example, verbal communication skills involve the proper understanding of a language, including itsphonology,orthography, syntax,lexicon, and semantics.[135]
Many aspects of human life depend on successful communication, from ensuring basic necessities of survival to building and maintaining relationships.[136]Communicative competence is a key factor regarding whether a person is able to reach their goals in social life, like having a successful career and finding a suitable spouse.[137]Because of this, it can have a large impact on the individual'swell-being.[138]The lack of communicative competence can cause problems both on the individual and the societal level, including professional, academic, and health problems.[139]
Barriers to effective communication can distort the message. They may result in failed communication and cause undesirable effects. This can happen if the message is poorly expressed because it uses terms with which the receiver is not familiar, or because it is not relevant to the receiver's needs, or because it contains too little or too much information. Distraction,selective perception, and lack of attention to feedback may also be responsible.[140]Noise is another negative factor. It concerns influences that interfere with the message on its way to the receiver and distort it.[141]Crackling sounds during a telephone call are one form of noise.Ambiguous expressionscan also inhibit effective communication and make it necessary todisambiguatebetween possible interpretations to discern the sender's intention.[142]These interpretations depend also on thecultural background of the participants. Significant cultural differences constitute an additional obstacle and make it more likely that messages are misinterpreted.[143]
Besides human communication, there are many other forms of communication found in the animal kingdom and among plants. They are studied in fields likebiocommunicationandbiosemiotics.[144]There are additional obstacles in this area for judging whether communication has taken place between two individuals. Acoustic signals are often easy to notice and analyze for scientists, but it is more difficult to judge whether tactile or chemical changes should be understood as communicative signals rather than as other biological processes.[145]
For this reason, researchers often use slightly altered definitions of communication to facilitate their work. A common assumption in this regard comes fromevolutionary biologyand holds that communication should somehow benefit the communicators in terms ofnatural selection.[146]The biologists Rumsaïs Blatrix and Veronika Mayer define communication as "the exchange of information between individuals, wherein both the signaller and receiver may expect to benefit from the exchange".[147]According to this view, the sender benefits by influencing the receiver's behavior and the receiver benefits by responding to the signal. These benefits should exist on average but not necessarily in every single case. This way, deceptive signaling can also be understood as a form of communication. One problem with the evolutionary approach is that it is often difficult to assess the impact of such behavior on natural selection.[148]Another common pragmatic constraint is to hold that it is necessary to observe a response by the receiver following the signal when judging whether communication has occurred.[149]
Animal communication is the process of giving and taking information among animals.[150]The field studying animal communication is calledzoosemiotics.[151]There are many parallels to human communication. One is that humans and many animals express sympathy by synchronizing their movements and postures.[152]Nonetheless, there are also significant differences, like the fact that humans also engage in verbal communication, which uses language, while animal communication is restricted to non-verbal (i.e. non-linguistic) communication.[153]Some theorists have tried to distinguish human from animal communication based on the claim that animal communication lacks areferential functionand is thus not able to refer to external phenomena. However, various observations seem to contradict this view, such as the warning signals in response to different types of predators used byvervet monkeys,Gunnison's prairie dogs, andred squirrels.[154]A further approach is to draw the distinction based on the complexity ofhuman language, especially its almost limitless ability to combine basic units of meaning into more complex meaning structures. One view states thatrecursionsets human language apart from all non-human communicative systems.[155]Another difference is that human communication is frequently linked to the conscious intention to send information, which is often not discernable for animal communication.[156]Despite these differences, some theorists use the term "animal language" to refer to certain communicative patterns in animal behavior that have similarities with human language.[157]
Animal communication can take a variety of forms, including visual, auditory, tactile,olfactory, and gustatory communication. Visual communication happens in the form of movements, gestures, facial expressions, and colors. Examples are movements seen duringmating rituals, the colors of birds, and the rhythmic light offireflies. Auditory communication takes place through vocalizations by species like birds,primates, and dogs. Auditory signals are frequently used to alert and warn. Lower-order living systems often have simple response patterns to auditory messages, reacting either by approach or avoidance.[158]More complex response patterns are observed for higher animals, which may use different signals for different types of predators and responses. For example, some primates use one set of signals for airborne predators and another for land predators.[159]Tactile communication occurs through touch,vibration, stroking, rubbing, and pressure. It is especially relevant for parent-young relations, courtship, social greetings, and defense. Olfactory and gustatory communication happen chemically through smells and tastes, respectively.[160]
There are large differences between species concerning what functions communication plays, how much it is realized, and the behavior used to communicate.[161]Common functions include the fields ofcourtshipand mating, parent-offspring relations, social relations, navigation, self-defense, andterritoriality.[162]One part of courtship and mating consists in identifying and attracting potential mates. This can happen through various means.Grasshoppersandcricketscommunicate acoustically by using songs,mothsrely on chemical means by releasingpheromones, and fireflies send visual messages by flashing light.[163]For some species, the offspring depends on the parent for its survival. One central function of parent-offspring communication is to recognize each other. In some cases, the parents are also able to guide the offspring's behavior.[164]
Social animals, likechimpanzees,bonobos, wolves, and dogs, engage in various forms of communication to express their feelings and build relations.[165]Communication can aid navigation by helping animals move through their environment in a purposeful way, e.g. to locate food, avoid enemies, and follow other animals. Inbats, this happens throughecholocation, i.e. by sending auditory signals and processing the information from the echoes.Beesare another often-discussed case in this respect since they perform a type ofdanceto indicate to other bees where flowers are located.[166]In regard to self-defense, communication is used to warn others and to assess whether a costly fight can be avoided.[167]Another function of communication is to mark and claim territories used for food and mating. For example, some male birds claim a hedge or part of a meadow by usingsongsto keep other males away and attract females.[168]
Two competing theories in the study of animal communication arenature theory and nurture theory. Their conflict concerns to what extent animal communication is programmed into the genes as a form of adaptation rather than learned from previous experience as a form ofconditioning.[169]To the degree that it is learned, it usually happens throughimprinting, i.e. as a form of learning that only occurs in a certain phase and is then mostly irreversible.[170]
Plant communicationrefers to plant processes involving the sending and receiving of information.[171]The field studying plant communication is calledphytosemiotics.[172]This field poses additional difficulties for researchers since plants are different from humans and other animals in that they lack acentral nervous systemand haverigid cell walls.[173]These walls restrict movement and usually prevent plants from sending and receiving signals that depend on rapid movement.[174]However, there are some similarities since plants face many of the same challenges as animals. For example, they need to find resources, avoid predators andpathogens, find mates, and ensure that their offspring survive.[175]Many of the evolutionary responses to these challenges are analogous to those in animals but are implemented using different means.[176]One crucial difference is thatchemical communicationis much more prominent in the plant kingdom in contrast to the importance of visual and auditory communication for animals.[177]
In plants, the termbehavioris usually not defined in terms of physical movement, as is the case for animals, but as a biochemical response to astimulus. This response has to be short relative to the plant's lifespan. Communication is a special form of behavior that involves conveying information from a sender to a receiver. It is distinguished from other types of behavior, like defensive reactions and mere sensing.[178]Like in the field of animal communication, plant communication researchers often require as additional criteria that there is some form of response in the receiver and that the communicative behavior is beneficial to sender and receiver.[179]Biologist Richard Karban distinguishes three steps of plant communication: the emission of a cue by a sender, the perception of the cue by a receiver, and the receiver's response.[180]For plant communication, it is not relevant to what extent the emission of a cue is intentional. However, it should be possible for the receiver to ignore the signal. This criterion can be used to distinguish a response to a signal from a defense mechanism against an unwanted change like intense heat.[181]
Plant communication happens in various forms. It includes communication within plants, i.e. withinplant cellsand between plant cells, between plants of the same or related species, and between plants and non-plant organisms, especially in theroot zone.[182]A prominent form of communication is airborne and happens throughvolatile organic compounds(VOCs). For example,mapletrees release VOCs when they are attacked by aherbivoreto warn neighboring plants, which then react accordingly by adjusting their defenses.[183]Another form of plant-to-plant communication happens throughmycorrhizal fungi. These fungi form underground networks, colloquially referred to as theWood-Wide Web, and connect the roots of different plants. The plants use the network to send messages to each other, specifically to warn other plants of a pest attack and to help prepare their defenses.[184]
Communication can also be observed for fungi and bacteria. Some fungal species communicate by releasingpheromonesinto the external environment. For instance, they are used to promote sexual interaction in several aquatic fungal species.[185]One form of communication between bacteria is calledquorum sensing. It happens by releasinghormone-like molecules, which other bacteria detect and respond to. This process is used to monitor the environment for other bacteria and to coordinate population-wide responses, for example, by sensing the density of bacteria and regulatinggene expressionaccordingly. Other possible responses include the induction ofbioluminescenceand the formation ofbiofilms.[186]
Most communication happens between members within a species as intraspecies communication. This is because the purpose of communication is usually some form of cooperation. Cooperation happens mostly within a species while different species are often in conflict with each other by competing over resources.[187]However, there are also some forms of interspecies communication.[188]This occurs especially forsymbioticrelations and significantly less forparasiticor predator-prey relations.[189]
Interspecies communication plays a key role for plants that depend on external agents for reproduction.[190]For example, flowers need insects forpollinationand provide resources likenectarand other rewards in return.[191]They use communication to signal their benefits and attract visitors by using distinctive colors and symmetrical shapes to stand out from their surroundings.[192]This form of advertisement is necessary since flowers compete with each other for visitors.[193]Many fruit-bearing plants rely on plant-to-animal communication to disperse their seeds and move them to a favorable location.[194]This happens by providing nutritious fruits to animals. The seeds are eaten together with the fruit and are later excreted at a different location.[195]Communication makes animals aware of where the fruits are and whether they are ripe. For many fruits, this happens through their color: they have an inconspicuous green color until they ripen and take on a new color that stands in visual contrast to the environment.[196]Another example of interspecies communication is found in the ant-plant relation.[197]It concerns, for instance, the selection of seeds byantsfor theirant gardensand the pruning of exogenous vegetation as well as plant protection by ants.[198]
Some animal species also engage in interspecies communication, like apes, whales, dolphins, elephants, and dogs.[199]For example, different species of monkeys use common signals to cooperate when threatened by a common predator.[200]Humans engage in interspecies communication when interacting withpetsandworking animals.[201]For instance, acoustic signals play a central role incommunication with dogs. Dogs can learn to react to various commands, like "sit" and "come". They can even be trained to respond to short syntactic combinations, like "bring X" or "put X in a box". They also react to the pitch and frequency of the human voice to detect emotions, dominance, and uncertainty. Dogs use a range of behavioral patterns to convey their emotions to humans, for example, in regard to aggressiveness, fearfulness, and playfulness.[202]
Computer communication concerns the exchange of data between computers and similar devices.[204]For this to be possible, the devices have to be connected through atransmission systemthat forms a network between them. Atransmitteris needed to send messages and a receiver is needed to receive them. A personal computer may use amodemas a transmitter to send information to a server through the public telephone network as the transmission system. The server may use a modem as its receiver.[205]To transmit the data, it has to be converted into an electric signal.[206]Communication channels used for transmission are eitheranalogordigitaland are characterized by features likebandwidthandlatency.[207]
There are many forms ofcomputer networks. The most commonly discussed ones areLANsandWANs.LANstands forlocal area network, which is a computer network within a limited area, usually with a distance of less than one kilometer.[208]This is the case when connecting two computers within a home or an office building. LANs can be set up using a wired connection, like Ethernet, or a wireless connection, likeWi-Fi.[209]WANs, on the other hand, arewide area networksthat span large geographical regions, like theinternet.[210]Their networks are more complex and may use several intermediate connection nodes to transfer information between endpoints.[211]Further types of computer networks includePANs(personal area networks),CANs(campus area networks), andMANs(metropolitan area networks).[212]
For computer communication to be successful, the involved devices have to follow a common set of conventions governing their exchange. These conventions are known as thecommunication protocol. They concern various aspects of the exchange, like the format of messages and how to respond to transmission errors. They also cover how the two systems are synchronized, for example, how the receiver identifies the start and end of a signal.[213]Based on the flow of informations, systems are categorized assimplex, half-duplex, and full-duplex. For simplex systems, signals flow only in one direction from the sender to the receiver, like in radio, cable television, and screens displaying arrivals and departures at airports.[214]Half-duplex systems allow two-way exchanges but signals can only flow in one direction at a time, likewalkie-talkiesandpolice radios. In the case of full-duplex systems, signals can flow in both directions at the same time, like regular telephone and internet.[215]In either case, it is often important for successful communication that the connection issecureto ensure that the transmitted data reaches only the intended destination and is not intercepted by an unauthorized third party.[216]This can be achieved by usingcryptography, which changes the format of the transmitted information to make it unintelligible to potential interceptors.[217]
Human-computer communication is a closely related field that concerns topics like howhumans interact with computersand how data in the form of inputs and outputs is exchanged.[218]This happens through auser interface, which includes the hardware used to interact with the computer, like amouse, akeyboard, and amonitor, as well as the software used in the process.[219]On the software side, most early user interfaces werecommand-line interfacesin which the user must type a command to interact with the computer.[220]Most modern user interfaces aregraphical user interfaces, likeMicrosoft WindowsandmacOS, which are usually much easier to use for non-experts. They involve graphical elements through which the user can interact with the computer, commonly using a design concept known asskeumorphismto make a new concept feel familiar and speed up understanding by mimicking the real-world equivalent of the interface object. Examples include the typical computer folder icon and recycle bin used for discarding files.[221]One aim when designing user interfaces is to simplify the interaction with computers. This helps make them more user-friendly and accessible to a wider audience while also increasing productivity.[222]
Communication studies, also referred to ascommunication science, is the academic discipline studying communication. It is closely related to semiotics, with one difference being that communication studies focuses more on technical questions of how messages are sent, received, and processed. Semiotics, on the other hand, tackles more abstract questions in relation tomeaningand how signs acquire it.[223]Communication studies covers a wide area overlapping with many other disciplines, such asbiology, anthropology,psychology,sociology,linguistics,media studies, andjournalism.[224]
Many contributions in the field of communication studies focus on developingmodelsandtheories of communication. Models of communication aim to give a simplified overview of the main components involved in communication. Theories of communication try to provide conceptual frameworks to accurately present communication in all its complexity.[225]Some theories focus on communication as a practical art of discourse while others explore the roles of signs, experience, information processing, and the goal of building a social order through coordinated interaction.[226]Communication studies is also interested in the functions and effects of communication. It covers issues like how communication satisfies physiological and psychological needs, helps build relationships, and assists in gathering information about the environment, other individuals, and oneself.[227]A further topic concerns the question of how communication systems change over time and how these changes correlate with other societal changes.[228]A related topic focuses on psychological principles underlying those changes and the effects they have on how people exchange ideas.[229]
Communication was studied as early asAncient Greece. Early influential theories were created byPlatoandAristotle, who stressed public speaking and the understanding ofrhetoric. According to Aristotle, for example, the goal of communication is to persuade the audience.[230]The field of communication studies only became a separate research discipline in the 20th century, especially starting in the 1940s.[231]The development of new communication technologies, such as telephone, radio, newspapers, television, and the internet, has had a big impact on communication and communication studies.[232]
Today, communication studies is a wide discipline. Some works in it try to provide a general characterization of communication in the widest sense. Others attempt to give a precise analysis of one specific form of communication. Communication studies includes many subfields. Some focus on wide topics like interpersonal communication, intrapersonal communication, verbal communication, and non-verbal communication. Others investigate communication within a specific area.[233]Organizational communicationconcerns communication between members of organizations such ascorporations,nonprofits, or small businesses. Central in this regard is the coordination of the behavior of the different members as well as the interaction with customers and the general public.[234]Closely related terms arebusiness communication,corporate communication, andprofessional communication.[235]The main element ofmarketing communicationisadvertisingbut it also encompasses other communication activities aimed at advancing the organization's objective to its audiences, likepublic relations.[236]Political communicationcovers topics likeelectoral campaignsto influence voters and legislative communication, like letters to acongressor committee documents. Specific emphasis is often given topropagandaand the role ofmass media.[237]
Intercultural communicationis relevant to both organizational and political communication since they often involve attempts to exchange messages between communicators from different cultural backgrounds.[238]The cultural background affects how messages are formulated and interpreted and can be the cause of misunderstandings.[239]It is also relevant fordevelopment communication, which is about the use of communication for assisting in development, like aid given byfirst-world countriestothird-world countries.[240]Health communicationconcerns communication in the field ofhealthcareand health promotion efforts. One of its topics is how healthcare providers, like doctors and nurses, should communicate with their patients.[241]
Communication history studies how communicative processes evolved and interacted with society, culture, and technology.[242]Human communication has a long history and the way people communicate has changed considerably over time. Many of these changes were triggered by the development of new communication technologies and had various effects on how people exchanged ideas.[243]New communication technologies usually require new skills that people need to learn to use them effectively.[244]
In the academic literature, the history of communication is usually divided into ages based on the dominant form of communication in that age. The number of ages and the precise periodization are disputed. They usually include ages for speaking, writing, and print as well as electronic mass communication and the internet.[245]According to communication theoristMarshall Poe, the dominant media for each age can be characterized in relation to several factors. They include the amount of information a medium can store, how long it persists, how much time it takes to transmit it, and how costly it is to use the medium. Poe argues that subsequent ages usually involve some form of improvement of one or more of the factors.[246]
According to some scientific estimates, language developed around 40,000 years ago while others consider it to be much older. Before this development, human communication resembled animal communication and happened through a combination of grunts, cries, gestures, and facial expressions. Language helped early humans to organize themselves and plan ahead more efficiently.[247]In early societies, spoken language was the primary form of communication.[248]Most knowledge was passed on through it, often in the form of stories or wise sayings. This form does not produce stable knowledge since it depends onimperfect human memory. Because of this, many details differ from one telling to the next and are presented differently by distinct storytellers.[249]As people started to settle and formagricultural communities, societies grew and there was an increased need for stable records of ownership of land and commercial transactions. This triggered the invention of writing, which is able to solve many problems that arose from using exclusively oral communication.[250]It is much more efficient at preserving knowledge and passing it on between generations since it does not depend on human memory.[251]Before the invention of writing, certain forms ofproto-writinghad already developed. Proto-writing encompasses long-lasting visible marks used to store information, like decorations on pottery items, knots in a cord to track goods, or seals to mark property.[252]
Most early written communication happened throughpictograms. Pictograms are graphical symbols that convey meaning by visually resembling real-world objects. The use of basic pictographic symbols to represent things like farming produce was common in ancient cultures and began around 9000 BCE. The first complex writing system including pictograms was developed around 3500 BCE by theSumeriansand is calledcuneiform.[253]Pictograms are still in use today, like no-smoking signs and the symbols of male and female figures on bathroom doors.[254]A significant disadvantage of pictographic writing systems is that they need a large amount of symbols to refer to all the objects one wants to talk about. This problem was solved by the development of other writing systems. For example, the symbols ofalphabeticwriting systems do not stand for regular objects. Instead, they relate to the sounds used in spoken language.[255]Other types of early writing systems includelogographicandideographicwriting systems.[256]A drawback of many early forms of writing, like the clay tablets used for cuneiform, was that they were not very portable. This made it difficult to transport the texts from one location to another to share information. This changed with the invention ofpapyrusby the Egyptians around 2500 BCE and was further improved later by the development ofparchmentandpaper.[257]
Until the 1400s, almost all written communication was hand-written, which limited the spread of written media within society since copying texts by hand was costly. The introduction and popularization of mass printing in the middle of the 15th century byJohann Gutenbergresulted in rapid changes. Mass printing quickly increased the circulation of written media and also led to the dissemination of new forms of written documents, like newspapers and pamphlets. One side effect was that the augmented availability of written documents significantly improved the generalliteracyof the population. This development served as the foundation for revolutions in various fields, including science, politics, and religion.[258]
Scientific discoveries in the 19th and 20th centuries caused many further developments in the history of communication. They include the invention oftelegraphsand telephones, which made it even easier and faster to transmit information from one location to another without the need to transport written documents.[259]These communication forms were initially limited to cable connections, which had to be established first. Later developments found ways of wireless transmission using radio signals. They made it possible to reach wide audiences and radio soon became one of the central forms of mass communication.[260]Various innovations in the field of photography enabled the recording of images on film, which led to the development of cinema and television.[261]The reach of wireless communication was further enhanced with the development ofsatellites, which made it possible to broadcast radio and television signals to stations all over the world. This way, information could be shared almost instantly everywhere around the globe.[262]The development of the internet constitutes a further milestone in the history of communication. It made it easier than ever before for people to exchange ideas, collaborate, and access information from anywhere in the world by using a variety of means, such as websites, e-mail, social media, and video conferences.[263]
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Trench codes(a form ofcryptography) werecodesused for secrecy by field armies inWorld War I.[1][2]Messages by field telephone, radio and carrier pigeons could be intercepted, hence the need for tacticalWorld War I cryptography. Originally, the most commonly used codes were simple substitution codes, but due to the relative vulnerability of theclassical cipher, trench codes came into existence. (Important messages generally used alternative encryption techniques for greater security.) The use of these codes required the distribution ofcodebooksto military personnel, which proved to be a security liability since these books could be stolen by enemy forces.[3]
By the middle ofWorld War Ithe conflict had settled down into a static battle of attrition, with the two sides sitting in huge lines of fixed earthwork fortifications. With armies generally immobile, distributing codebooks and protecting them was easier than it would have been for armies on the move. However, armies were still in danger of trench-raiding parties who would sneak into enemy lines and try to snatch codebooks. When this happened, an alarm could be raised and a code quickly changed. Trench codes were changed on a regular basis in an attempt to prevent code breakers from deciphering messages.[1]
TheFrench Armybegan to develop trench codes in early 1916. They started astelephonecodes, implemented at the request of ageneralwhose forces had suffered devastatingartillery barragesdue to indiscretions in telephone conversations between his men. The original telephone code featured a small set of two-letter codewords that were spelled out in voice communications. This grew into a three-letter code scheme, soon adopted for wireless, with early one-part code implementations evolving into more secure two-part code implementations. TheBritishbegan to adopt trench codes as well.
TheImperial German Armystarted using trench codes in the spring of 1917, evolving into a book of 4,000 codewords that were changed twice a month, with different codebooks used on different sectors of the front. French codebreakers were extremely competent at crackingciphersbut were somewhat inexperienced at cracking codes, which require a slightly different mindset. It took them time to get to the point where they were able to crack the German codes in a timely fashion.
The Americans were relative newcomers to cryptography when they entered the war, but they did have their star players. One wasParker Hitt, b. 1878, who before the war had been anArmy Signal Corpsinstructor. He was one of the first to try to bringUS Armycryptology into the 20th century, publishing an influential short work on the subject in 1915 called theManual for the solution of military ciphers.[4]He was assigned to France in an administrative role, but his advice was eagerly sought by colleagues working in operational cryptology. Another Signal Corps officer who would make his mark on cryptology wasJoseph Mauborgne, who in 1914, as afirst lieutenant, had been the first to publish a solution to thePlayfair cipher.
When the Americans began moving up to the front in numbers in early 1918, they adopted trench codes[1]: p. 222and became very competent at their construction, with aCaptainHoward R. Barneseventually learning to produce them at a rate that surprised British colleagues. The Americans adopted a series of codes named after rivers, beginning with "Potomac". They learned to print the codebooks on paper that burned easily and degraded quickly after a few weeks, when the codes would presumably be obsolete, while using a typeface that was easy to read under trench conditions.
American code makers were often frustrated by the inability or refusal of combat units to use the codes—or worse, to use them properly. Lt. Col. Frank Moorman, reviewing U.S wireless intelligence in 1920. wrote:
That will be the real problem for the future, to make the men at the front realize the importance of handling codes carefully and observing "foolish" little details that the code man insists on. They cannot see the need of it and they do not want to do it. They will do anything they can to get out of it. My idea would be to hang a few of the offenders. This would not only get rid of some but would discourage the development of others. It would be a saving of lives to do it. It is a sacrifice of American lives to unnecessarily assist the enemy in the solution of our code.[1]: p.269
Below are pages from a U.S. Army World War I trench code, an edition designated as "Seneca:"[1]: pp.185–188
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The vulnerability ofJapanese naval codesand ciphers was crucial to the conduct ofWorld War II, and had an important influence on foreign relations between Japan and the west in the years leading up to the war as well. Every Japanese code was eventually broken, and the intelligence gathered made possible such operations as the victorious American ambush of the Japanese Navy atMidwayin 1942 (by breaking code JN-25b) and the shooting down of Japanese admiralIsoroku Yamamotoa year later inOperation Vengeance.
TheImperial Japanese Navy(IJN) used manycodesandciphers. All of these cryptosystems were known differently by different organizations; the names listed below are those given by Western cryptanalytic operations.
The Red Book code was an IJNcode booksystem used inWorld War Iand after. It was called "Red Book" because the American photographs made of it were bound in red covers.[1]It should not be confused with theRED cipherused by the diplomatic corps.
This code consisted of two books. The first contained the code itself; the second contained an additive cipher which was applied to the codes before transmission, with the starting point for the latter being embedded in the transmitted message. A copy of the code book was obtained in a"black bag" operationon the luggage of a Japanese naval attaché in 1923; after three years of workAgnes Driscollwas able to break the additive portion of the code.[2][3][4]
Knowledge of the Red Book code helped crack the similarly constructed Blue Book code.[1]
A cipher machine developed for Japanese naval attaché ciphers, similar to JADE. It was not used extensively,[5][6]but Vice AdmiralKatsuo Abe, a Japanese representative to the Axis Tripartite Military Commission, passed considerable information about German deployments in CORAL, intelligence "essential for Allied military decision making in the European Theater."[7]
A cipher machine used by the Imperial Japanese Navy from late 1942 to 1944 and similar to CORAL.
A succession of codes used to communicate between Japanese naval installations. These were comparatively easily broken by British codebreakers in Singapore and are believed to have been the source of early indications of imminent naval war preparations.[8]
The Fleet Auxiliary System, derived from theJN-40merchant-shipping code. Important for information on troop convoys and orders of battle.
An inter-island cipher that provided valuable intelligence, especially when periodic changes to JN-25 temporarily blacked out U.S. decryption. JN-20 exploitation produced the "AF is short of water" message that established the main target of the Japanese Fleet, leading to a decisive U.S. victory at theBattle of Midwayin 1942.[9]: p.155
JN-25is the name given by codebreakers to the main, and most secure, command and control communications scheme used by the IJN during World War II.[10]Named as the 25th Japanese Navy system identified, it was initially given the designation AN-1 as a "research project" rather than a "current decryption" job. The project required reconstructing the meaning of thirty thousand code groups and piecing together thirty thousand random additives.[11]
Introduced from 1 June 1939 to replace Blue (and the most recent descendant of the Red code),[12]it was an enciphered code, producing five-numeral groups for transmission. New code books andsuper-encipheringbooks were introduced from time to time, each new version requiring a more or less fresh cryptanalytic attack.John Tiltmanwith some help fromAlan Turing(at GCSB,Government Communications Security Bureau) had "solved" JN25 by 1941, i.e. they knew that it was a five-digit code with a codebook to translate words into five digits and there was a second "additive" book that the sender used to add to the original numbers "But knowing all this didn’t help them read a single message".
By April 1942 JN25 was about 20 percent readable, so codebreakers could read "about one in five words" andtraffic analysiswas far more useful.[13]Tiltman had devised a (slow; neither easy nor quick) method of breaking it and had noted that all the numbers in the codebook were divisible by three.[14]"Breaking" rather than "solving" a code involves learning enough code words and indicators so that any given message can be read.[15]
In particular, JN-25 was significantly changed on 1 December 1940 (JN25a);[12]and again on 4 December 1941 (JN25b),[16]just before theattack on Pearl Harbor.
British, Australian, Dutch and American cryptanalysts co-operated on breaking JN-25 well before the Pearl Harbor attack, but because the Japanese Navy was not engaged in significant battle operations before then, there was little traffic available to use as raw material. Before then, IJN discussions and orders could generally travel by routes more secure than broadcast, such as courier or direct delivery by an IJN vessel. Publicly available accounts differ, but the most credible agree that the JN-25 version in use before December 1941 was not more than perhaps 10% broken at the time of the attack,[17]and that primarily in stripping away its super-encipherment. JN-25 traffic increased immensely with the outbreak of naval warfare at the end of 1941 and provided the cryptographic "depth" needed to succeed in substantially breaking the existing and subsequent versions of JN-25.
The American effort was directed from Washington, D.C. by the U.S. Navy's signals intelligence command,OP-20-G; at Pearl Harbor it was centered at the Navy's Combat Intelligence Unit (Station HYPO, also known as COM 14),[18]led by CommanderJoseph Rochefort.[10]However, in 1942 not every cryptogram was decoded, as Japanese traffic was too heavy for the undermanned Combat Intelligence Unit.[19]With the assistance ofStation CAST(also known as COM 16, jointly commanded by Lts Rudolph Fabian and John Lietwiler)[20]in the Philippines, and the BritishFar East Combined Bureauin Singapore, and using apunched cardtabulating machinemanufactured byInternational Business Machines, a successful attack was mounted against the 4 December 1941 edition (JN25b). Together they made considerable progress by early 1942."Cribs"exploited common formalities in Japanese messages, such as "I have the honor to inform your excellency" (seeknown plaintext attack).
Later versions of JN-25 were introduced: JN-25c from 28 May 1942, deferred from 1 April then 1 May; providing details of the attacks on Midway and Port Moresby. JN-25d was introduced from 1 April 1943, and while the additive had been changed, large portions had been recovered two weeks later, which provided details of Yamamoto's plans that were used inOperation Vengeance, the shooting-down of his plane.[21]
This was a naval code used by merchant ships (commonly known as the "marucode"),[22]broken in May 1940. 28 May 1941, when thewhalefactory shipNisshin Maru No. 2 (1937)visited San Francisco,U.S. Customs ServiceAgent George Muller and Commander R. P. McCullough of the U.S. Navy's12th Naval District(responsible for the area) boarded her and seized her codebooks, without informingOffice of Naval Intelligence(ONI). Copies were made, clumsily, and the originals returned.[23]The Japanese quickly realized JN-39 was compromised, and replaced it with JN-40.[24]
JN-40 was originally believed to be a code super-enciphered with a numerical additive, in the same way as JN-25. However, in September 1942, an error by the Japanese gave clues to John MacInnes and Brian Townend, codebreakers at the BritishFECB,Kilindini. It was afractionating transposition cipherbased on a substitution table of 100 groups of two figures each followed by acolumnar transposition. By November 1942, they were able to read all previous traffic and break each message as they received it. Enemy shipping, including troop convoys, was thus trackable, exposing it to Allied attack. Over the next two weeks they broke two more systems, the "previously impenetrable" JN167 and JN152.[24][25]
The "minor operations code" often contained useful information on minor troop movements.[26]
A simple transposition and substitution cipher used for broadcasting navigation warnings. In 1942 after breaking JN-40 the FECB at Kilindini broke JN-152 and the previously impenetrable JN-167, another merchant shipping cypher.[27][28]
A merchant-shipping cipher (see JN-152).
In June 1942 theChicago Tribune, run byisolationistCol. Robert R. McCormick, published an article implying that the United States had broken the Japanese codes, saying the U.S. Navy knew in advance about the Japanese attack on Midway Island, and published dispositions of the Japanese invasion fleet. The executive officer ofLexington, CommanderMorton T. Seligman(who was transferred to shore duties), had shown Nimitz's executive order to reporterStanley Johnston.
The government at first wanted to prosecute theTribuneunder theEspionage Act of 1917. For various reasons, including the desire not tobring more attentionto the article and because the Espionage Act did not coverenemysecrets, the charges were dropped. A grand jury investigation did not result in prosecution but generated further publicity and, according toWalter Winchell, "tossed security out of the window". Several in Britain believed that their worst fears about American security were realized.[29]
In early August, a RAN intercept unit in Melbourne (FRUMEL) heard Japanese messages, using a superseded lower-grade code. Changes were made to codebooks and the call-sign system, starting with the new JN-25c codebook (issued two months before). However the changes indicated the Japanese believed the Allies had worked out the fleet details from traffic analysis or had obtained a codebook and additive tables, being reluctant to believe that anyone could have broken their codes (least of all a Westerner).[30]
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see also
TheZimmermann telegram(orZimmermann noteorZimmermann cable) was a secret diplomatic communication issued from theGerman Foreign Officeon January 17, 1917, that proposed a military contract between theGerman EmpireandMexicoif theUnited StatesenteredWorld War Iagainst Germany. With Germany's aid, Mexicowould recoverTexas,Arizona, andNew Mexico. The telegram was intercepted byBritish intelligence.
Revelation of the contents enraged Americans, especially after German State Secretary for Foreign AffairsArthur Zimmermannpublicly admitted on March 3, 1917, that the telegram was genuine. It helped to generate support for theAmerican declaration of war on Germanyin April 1917.[1]
The decryption has been described as the most significant intelligence triumph for Britain during World War I[2]and it marked one of the earliest occasions on which a piece ofsignals intelligenceinfluenced world events.[3]The decryption was possible after the failure of theNiedermayer-Hentig ExpeditiontoAfghanistan, whenWilhelm Wassmussabandoned his codebook, which the Allies later recovered, and allowed the British to decrypt the Zimmermann telegram.[4]
The message came in the form of acodedtelegram dispatched byArthur Zimmermann, theStaatssekretär(a top-levelcivil servant, second only to their respective minister) in the Foreign Office of the German Empire on January 17, 1917. The message was sent to the German ambassador to Mexico,Heinrich von Eckardt.[5]Zimmermann sent the telegram in anticipation of the resumption ofunrestricted submarine warfareby Germany on February 1, which the German government presumed would almost certainly lead to war with the United States. The telegram instructed Von Eckardt that if the United States appeared certain to enter the war, he was to approach the Mexican government with a proposal for military alliance with funding from Germany. The decoded telegram was as follows:[6]
Original (German):
Wir beabsichtigen am 1. Februar uneingeschränkten Ubootkrieg zu beginnen. Es wird Versucht werden, Amerika trotzdem neutral zu halten.
Für den Fall, daß dies nicht gelingen sollte, schlagen wir Mexico mit folgender Grundlage Bündnis vor; Gemeinsame Kriegführung, gemeinsamer Friedensschluß. Reichliche finzanzielle Unterstützung und Einverständnis unsererseits, daß Mexiko in Texas, Neu Mexiko, Arizona früher verlorenes Gebiet zurückerobert. Regelung im einzelnen Euer Hochwohlgeboren überlassen.
Euer pp. wollen Vorstehendes Präsidenten streng geheim eröffnen, sobald Kriegsausbruch mit Vereinigten Staaten feststeht und Anregung hinzufügen, Japan von sich aus zu fortigem Beitritt einzuladen und gleichzeitig zwischen uns und Japan zu vermitteln.
Bitte Präsidenten darauf hinweisen, daß rücksichtslose Anwendung unserer U-boote jetzt Aussicht bietet, England in wenigen Monaten sum Frieden zu zwingen.Empfang bestätigen.
Zimmerman.[7]
Translated:
On February 1 we intend to begin submarine
warfare without restriction. In spite of this it is our
intention to endeavour to keep the United States
neutral. If this attempt is not successful, we propose
an alliance on the following basis with Mexico:
That we shall make war together and together
make peace; we shall give general financial support,
and it is understood that Mexico is to reconquer
her lost territory of New Mexico, Texas and
Arizona. The details are left to you for settlement.
You are instructed to inform the President of
Mexico of the above in the greatest confidence as
soon as it is certain that there will be an outbreak
of war with the United States, and suggest that the
President of Mexico shall on his own initiative communicate
with Japan suggesting the latter's adherence
at once to this plan, and at the same time
offer to mediate between Germany and Japan.
Please call to the attention of the President of
Mexico that the employment of ruthless submarine
warfare now promises to compel England to make
peace in a few month [sic]. – Zimmerman.[7]
(The signature dropped the secondnof the nameZimmermannfor telegraphic purposes.[7])
Germany had long sought to incite a war between Mexico and the United States, which would have tied down American forces and slowed the export of American arms to theAllies.[8]The Germans had aided in arming Mexico, as shown by the 1914Ypiranga incident.[9]German Naval IntelligenceofficerFranz von Rintelenhad attempted to incite a war between Mexico and the United States in 1915, givingVictoriano Huerta$12 million for that purpose.[10]The German saboteurLothar Witzke, who was based in Mexico City, claimed to be responsible for the March 1917 munitions explosion at theMare Island Naval Shipyardin the San Francisco Bay Area,[11]and was possibly responsible for the July 1916Black Tom explosionin New Jersey.
The failure of United States troopsto capture Pancho Villa in 1916and the movement of President Carranza in favor of Germany emboldened the Germans to send the Zimmermann note.[12]
The German provocations were partially successful. PresidentWoodrow Wilsonordered themilitary invasion of Veracruzin 1914 in the context of the Ypiranga incident and against the advice of the British government.[13]War was prevented thanks to theNiagara Falls peace conferenceorganized by theABC nations, but the occupation was a decisive factor inMexican neutrality in World War I.[14]Mexico refused to participate in the embargo against Germany and granted full guarantees to the German companies for keeping their operations open, specifically in Mexico City.[15]
The Zimmermann telegram was part of an effort carried out by the Germans to postpone the transportation of supplies and other war materials from the United States to the Allies, which were at war against Germany.[17]The main purpose of the telegram was to make the Mexican government declare war on the United States in hopes of tying down American forces and slowing the export of American arms.[18]TheGerman High Commandbelieved that it could defeat the British and French on theWestern Frontand strangle Britain with unrestricted submarine warfare before American forces could be trained and shipped to Europe in sufficient numbers to aid the Allies. The Germans were encouraged by their successes on theEastern Frontto believe that they could divert large numbers of troops to the Western Front in support of their goals.[citation needed]
Mexican PresidentVenustiano Carranzaassigned a military commission to assess the feasibility of the Mexican takeover of their former territories contemplated by Germany.[19]The generals concluded that such a war was unwinnable for the following reasons:
The Carranza government was recognizedde jureby the United States on August 31, 1917, as a direct consequence of the Zimmermann telegram to ensureMexican neutrality during World War I.[21][22]After themilitary invasion of Veracruzin 1914, Mexico did not participate in any military excursion with the United States in World War I.[14]That ensured that Mexican neutrality was the best outcome that the United States could hope for even if it allowed German companies to keep their operations in Mexico open.[15]
The decryption was possible after the failure of theNiedermayer-Hentig ExpeditiontoAfghanistan, whenWilhelm Wassmussabandoned his codebook, which the Allies later recovered, and allowed the British to decrypt the Zimmermann telegram.[23]
Zimmermann's office sent the telegram to the German embassy in the United States for retransmission to Von Eckardt in Mexico. It has traditionally been understood that the telegram was sent over three routes. It went by radio, and passed via telegraph cable inside messages sent by diplomats of two neutral countries (the United States and Sweden).
Direct telegraph transmission of the telegram was impossible because the British had cut the German international cables at the outbreak of war. However, Germany could communicate wirelessly through the Telefunken plant, operating under Atlantic Communication Company inWest Sayville, New York, where the telegram was relayed to the Mexican Consulate. Ironically, the station was under the control of theUS Navy, which operated it for Atlantic Communication Company, the American subsidiary of the German entity.
The Swedish diplomatic message holding the Zimmerman telegram went from Stockholm to Buenos Aires over British submarine telegraph cables, and then moved from Buenos Aires to Mexico over the cable network of a United States company.
After the Germans' telegraph cables had been cut, the German Foreign Office appealed to the United States for use of their diplomatic telegraphic messages for peace messages. President Wilson agreed in the belief both that such co-operation would sustain continued good relations with Germany and that more efficient German–American diplomacy could assist Wilson's goal of a negotiated end to the war. The Germans handed in messages to the American embassy in Berlin, which were relayed to the embassy in Denmark and then to the United States by American telegraph operators. The Germans assumed that this route was secure and so used it extensively.[24]
However, that put German diplomats in a precarious situation since they relied on the United States to transmit Zimmermann's note to its final destination, but the message's unencrypted contents would be deeply alarming to the Americans. The United States had placed conditions on German usage, most notably that all messages had to be in cleartext (uncoded). However, Wilson had later reversed the order and relaxed the wireless rules to allow coded messages to be sent.[25]Thus the Germans were able to persuadeUS AmbassadorJames W. Gerardto accept Zimmermann's note in coded form, and it was transmitted on January 16, 1917.[24]
All traffic passing through British hands came toBritish intelligence, particularly to thecodebreakersand analysts inRoom 40at theAdmiralty.[24]In Room 40,Nigel de Greyhad partially decoded the telegram by the next day.[26]By 1917, the diplomatic code 13040 had been in use for many years. Since there had been ample time for Room 40 to reconstruct the code cryptanalytically, it was readable to a fair degree. Room 40 had obtained German cryptographic documents, including the diplomatic code 3512 (captured during theMesopotamian campaign), which was a later updated code that was similar to but not really related to code 13040, and naval code SKM (Signalbuch der Kaiserlichen Marine), which was useless for decoding the Zimmermann telegram but valuable to decode naval traffic, which had been retrieved from the wrecked cruiserSMSMagdeburgby the Russians, who passed it to the British.[27]
Disclosure of the telegram would sway American public opinion against Germany if the British could convince the Americans that the text was genuine, but the Room 40 chief William Reginald Hall was reluctant to let it out because the disclosure would expose the German codes broken in Room 40 and British eavesdropping on United States diplomatic traffic. Hall waited three weeks during which de Grey and cryptographerWilliam Montgomerycompleted the decryption. On February 1, Germany announced resumption of "unrestricted" submarine warfare, an act that led the United States to break off diplomatic relations with Germany on February 3.[24]
Hall passed the telegram to the British Foreign Office on February 5 but still warned against releasing it. Meanwhile, the British discussed possible cover stories to explain to the Americans how they obtained the coded text of the telegram and to explain how they obtained the cleartext of the telegram without letting anyone know that the codes had been broken. Furthermore, the British needed to find a way to convince the Americans the message was not a forgery.[28]
For the first story, the British obtained the coded text of the telegram from the Mexican commercial telegraph office. The British knew that since the German embassy in Washington would relay the message by commercial telegraph, the Mexican telegraph office would have the coded text. "Mr. H", a British agent in Mexico, bribed an employee of the commercial telegraph company for a copy of the message. SirThomas Hohler, the British ambassador in Mexico, later claimed to have been "Mr. H" or at least to have been involved with the interception in his autobiography.[29]The coded text could then be shown to the Americans without embarrassment.
Moreover, the retransmission was encoded with the older code 13040 and so by mid-February, the British had the complete text and the ability to release the telegram without revealing the extent to which the latest German codes had been broken. (At worst, the Germans might have realized that the 13040 code had been compromised, but that was a risk worth taking against the possibility of United States entry into the war.) Finally, since copies of the 13040 code text would also have been deposited in the records of the American commercial telegraph company, the British had the ability to prove the authenticity of the message to the American government.[3]
As a cover story, the British could publicly claim that their agents had stolen the telegram's decoded text in Mexico. Privately, the British needed to give the Americans the 13040 code so that the American government could verify the authenticity of the message independently with their own commercial telegraphic records, but the Americans agreed to back the official cover story. The German Foreign Office refused to consider that their codes could have been broken but sent Von Eckardt on a witch hunt for a traitor in the embassy in Mexico. Von Eckardt indignantly rejected those accusations, and the Foreign Office eventually declared the embassy exonerated.[24]
On February 19, Hall showed the telegram to Edward Bell, the secretary of the American Embassy in Britain. Bell was at first incredulous and thought that it was a forgery. Once Bell was convinced the message was genuine, he became enraged. On February 20, Hall informally sent a copy to US AmbassadorWalter Hines Page. On February 23, Page met with British Foreign MinisterArthur Balfourand was given the codetext, the message in German, and the English translation. The British had obtained a further copy in Mexico City, and Balfour could obscure the real source with the half-truth that it had been "bought in Mexico".[30]Page then reported the story to Wilson on February 24, 1917, including details to be verified from telegraph-company files in the United States. Wilson felt "much indignation" toward the Germans and wanted to publish the Zimmermann Telegraph immediately after he had received it from the British, but he delayed until March 1, 1917.[31]
Many Americans then heldanti-Mexicanas well asanti-Germanviews. Mexicans had a considerable amount ofanti-American sentimentin return, some of which was caused by theAmerican occupation of Veracruz.[32]GeneralJohn J. Pershinghad long been chasingthe revolutionaryPancho Villafor raiding into American territory and carried out several cross-border expeditions. News of the telegram further inflamed tensions between the United States and Mexico.
However, many Americans, particularly those withGermanorIrishancestry, wished to avoid the conflict in Europe. Since the public had been told falsely that the telegram had been stolen in a decoded form in Mexico, the message was at first widely believed to be an elaborate forgery created by British intelligence. That belief, which was not restricted to pacifist and pro-German lobbies, was promoted by German and Mexican diplomats alongside some antiwar American newspapers, especially those of theHearstpress empire.
On February 1, 1917, Germany had begun unrestricted submarine warfare against all ships in the Atlantic bearing the American flag, both passenger and merchant ships. Two ships were sunk in February, and most American shipping companies held their ships in port. Besides the highly-provocative war proposal to Mexico, the telegram also mentioned "ruthless employment of our submarines". Public opinion demanded action. Wilson had refused to assign US Navy crews and guns to the merchant ships, but once the Zimmermann note was public, Wilson called for arming the merchant ships although antiwar members of theUS Senateblocked his proposal.[33]
TheWilson administrationnevertheless remained with a dilemma. Evidence the United States had been provided confidentially by the British informed Wilson that the message was genuine, but he could not make the evidence public without compromising the British codebreaking operation. This problem was, however, resolved when any doubts as to the authenticity of the telegram were removed by Zimmermann himself. At a press conference on March 3, 1917, he told an American journalist, "I cannot deny it. It is true." Then, on March 29, 1917, Zimmermann gave a speech in theReichstagin which he admitted that the telegram was genuine.[34]Zimmermann hoped that Americans would understand that the idea was that Germany would not fund Mexico's war with the United States unless the Americans joined World War I. Nevertheless, in his speech Zimmermann questioned how the Washington government obtained the telegram.[35]According toReutersnews service, Zimmermann told the Reichstag, "the instructions...came into its hands in a way which was not unobjectionable."[35]
On April 6, 1917, Congress voted todeclare war on Germany. Wilson had askedCongressfor "awar to end all wars" that would "make the world safe for democracy".[36]
Wilson considered another military invasion of Veracruz andTampicoin 1917–1918,[37][38]to pacify theIsthmus of Tehuantepecand Tampico oil fields and to ensure their continued production during the civil war,[38][39]but this time, Mexican PresidentVenustiano Carranza, recently installed, threatened to destroy the oil fields if theUS Marineslanded there.[40][41]
The Japanese government, another nation mentioned in the Zimmerman telegram, was alreadyinvolved in World War I, on the side of the Allies against Germany. The government later released a statement that Japan was not interested in changing sides or attacking America.[42][43]
In October 2005, it was reported that an original typescript of the decoded Zimmermann telegram had recently been discovered by an unnamed historian who was researching and preparing a history of the United Kingdom'sGovernment Communications Headquarters. The document is believed to be the actual telegram shown to the American ambassador in London in 1917. Marked in Admiral Hall's handwriting at the top of the document are the words: "This is the one handed to Dr Page and exposed by the President." Since many of the secret documents in this incident had been destroyed, it had previously been assumed that the original typed "decrypt" was gone forever. However, after the discovery of this document, the GCHQ official historian said: "I believe that this is indeed the same document that Balfour handed to Page."[44]
As of 2006, there were six "closed" files on the Zimmermann telegram which had not been declassified held byThe National Archivesat Kew (formerly thePRO).[45]
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Acode talkerwas a person employed by the military during wartime to use a little-known language as a means of secret communication. The term is most often used for United States service members during theWorld Warswho used their knowledge ofNative American languagesas a basis to transmit coded messages. In particular, there were approximately 400 to 500Native Americansin theUnited States Marine Corpswhose primary job was to transmit secrettacticalmessages. Code talkers transmitted messages over military telephone or radio communications nets using formally or informally developed codes built upon their indigenous languages. The code talkers improved the speed ofencryptionand decryption of communications infront lineoperations duringWorld War IIand are credited with some decisive victories. Their code was never broken.
There were two code types used during World War II. Type one codes were formally developed based on the languages of theComanche,Hopi,Meskwaki, andNavajopeoples. They used words from their languages for each letter of the English alphabet. Messages could be encoded and decoded by using asimple substitution cipherwhere theciphertextwas the Native language word. Type two code was informal and directly translated from English into the Indigenous language. Code talkers used short, descriptive phrases if there was no corresponding word in the Indigenous language for the military word. For example, the Navajo did not have a word forsubmarine, so they translated it asiron fish.[1][2]
The termCode Talkerwas originally coined by the United States Marine Corps and used to identify individuals who completed the special training required to qualify as Code Talkers. Their service records indicated "642 – Code Talker" as a duty assignment. Today, the term Code Talker is still strongly associated with the bilingualNavajospeakers trained in the Navajo Code during World War II by the US Marine Corps to serve in all sixdivisions of the Corpsand theMarine Raidersof thePacific theater. However, the use of Native American communicators pre-dates WWII. Early pioneers of Native American-based communications used by the US Military include theCherokee,Choctaw, andLakota peoplesduring World War I.[3]Today the term Code Talker includes military personnel from all Native American communities who have contributed their language skills in service to the United States.
Other Native American communicators—now referred to as code talkers—were deployed by theUnited States Armyduring World War II, includingLakota,[4]Meskwaki,Mohawk,[5][6]Comanche,Tlingit,[7]Hopi,[8]Cree, andCrowsoldiers; they served in the Pacific, North African, and European theaters.[9]
Native speakers of theAssiniboine languageserved as code talkers during World War II to encrypt communications.[10]One of these code talkers wasGilbert Horn Sr., who grew up in theFort Belknap Indian Reservationof Montana and became a tribal judge and politician.[10]
In November 1952,Euzko Deyamagazine[11]reported that sometime in May 1942, upon meeting a large number of US Marines ofBasqueancestry in a San Francisco camp, CaptainFrank D. Carranzahad thought of using theBasque languagefor codes.[12][13][14]His superiors were concerned about risk, as there were known settlements of Basque people in the Pacific region, including 35 BasqueJesuitsinHiroshima, led byPedro Arrupe; a colony of Basquejai alaiplayers in China and the Philippines; and Basque supporters ofFalangein Asia. Consequently, the US Basque code talkers were not deployed in these theaters; instead, they were used initially in tests and in transmitting logistics information for Hawaii and Australia.
According toEuzko Deya, on August 1, 1942, Lieutenants Nemesio Aguirre, Fernández Bakaicoa, and Juanana received a Basque-coded message from San Diego for AdmiralChester Nimitz. The message warned Nimitz ofOperation Appleto remove the Japanese from theSolomon Islands. They also translated the start date, August 7, forthe attack on Guadalcanal. As the war extended over the Pacific, there was a shortage of Basque speakers, and the US military came to prefer the parallel program based on the use of Navajo speakers.
In 2017, Pedro Oiarzabal and Guillermo Tabernilla published a paper refutingEuzko Deya's article.[15]According to Oiarzabal and Tabernilla, they could not find Carranza, Aguirre, Fernández Bakaicoa, or Juanana in theNational Archives and Records Administrationor US Army archives. They did find a small number of US Marines withBasque surnames, but none of them worked in transmissions. They suggest that Carranza's story was anOffice of Strategic Servicesoperation to raise sympathy for US intelligence among Basque nationalists.
The US military's first known use of code talkers was during World War I.Cherokeesoldiers of the US30th Infantry Divisionfluent in theCherokee languagewere assigned to transmit messages while under fire during theSecond Battle of the Somme. According to the Division Signal Officer, this took place in September 1918 when their unit was under British command.[16][17]
DuringWorld War I, company commander Captain Lawrence of the US Army overheard Solomon Louis and Mitchell Bobb having a conversation inChoctaw. Upon further investigation, he found eightChoctawmen served in the battalion. The Choctaw men in the Army's36th Infantry Divisionwere trained to use their language in code. They helped theAmerican Expeditionary Forcesin several battles of theMeuse-Argonne Offensive. On October 26, 1918, the code talkers were pressed into service and the "tide of battle turned within 24 hours ... and within 72 hours the Allies were on full attack."[18][19]
German authorities knew about the use of code talkers during World War I. Germans sent a team of thirtyanthropologiststo the United States to learn Native American languages before the outbreak of World War II.[20][21]However, the task proved too difficult because of the large array of Indigenous languages anddialects. Nonetheless, after learning of the Nazi effort, the US Army opted not to implement a large-scale code talker program in theEuropean theater.
Initially, 17 code talkers were enlisted, but three could not make the trip across the Atlantic until the unit was finally deployed.[22]A total of 14 code talkers using theComanche languagetook part in theInvasion of Normandyand served in the4th Infantry Divisionin Europe.[23]Comanche soldiers of the 4th Signal Company compiled a vocabulary of 250 code terms using words and phrases in their own language.[24]Using a substitution method similar to that of theNavajo, the code talkers used descriptive words from the Comanche language for things that did not have translations. For example, the Comanche language code term fortankwasturtle,bomberwaspregnant bird,machine gunwassewing machine, andAdolf Hitlerwascrazy white man.[25][26]
Two Comanche code talkers were assigned to each regiment, and the remainder were assigned to the 4th Infantry Division headquarters. The Comanche began transmitting messages shortly after landing onUtah Beachon June 6, 1944. Some were wounded but none killed.[25]
In 1989, the French government awarded the Comanche code talkers theChevalierof theNational Order of Merit. On November 30, 1999, theUnited States Department of DefensepresentedCharles Chibittywith theKnowlton Award, in recognition of his outstanding intelligence work.[25][27]
InWorld War II, theCanadian Armed Forcesemployed First Nations soldiers who spoke theCree languageas code talkers. Owing to oaths of secrecy and official classification through 1963, the role of Cree code talkers was less well-known than their US counterparts and went unacknowledged by the Canadian government.[28]A 2016 documentary,Cree Code Talkers, tells the story of one suchMétisindividual,Charles "Checker" Tomkins. Tomkins died in 2003 but was interviewed shortly before his death by the SmithsonianNational Museum of the American Indian. While he identified other Cree code talkers, "Tomkins may have been the last of his comrades to know anything of this secret operation."[29][30]
In 2022 during theRusso-Ukrainian War, theHungarian languageis reported to be used by theUkrainian armyto relay operational military information and orders to circumvent being understood by the invadingRussian armywithout the need to encrypt and decipher the messages.[31][32]Ukraine has a sizeableHungarian populationof over 150,000 people who live mainly in theKárpátalja (in Hungarian) or Zakarpatska Oblast (in Ukrainian) divisionof Ukraine, adjacent toHungary. As Ukrainian nationals, men of enlistment age are also subject to military service, hence theUkrainian armyhas a Hungarian-speaking capability. It is one of the most spoken and official languages of thisregion in present-day Ukraine. TheHungarian languageis not anIndo-European languagelike theSlavicUkrainianorRussian, but aUralic language. For this reason, it is distinct and incomprehensible for Russian speakers.[citation needed]
A group of 27Meskwakienlisted in the US Army together in January 1941; they comprised 16 percent of Iowa's Meskwaki population. During World War II, the US Army trained eight Meskwaki men to use their nativeFox languageas code talkers. They were assigned to North Africa. The eight were posthumously awarded theCongressional Gold Medalin 2013; the government gave the awards to representatives of the Meskwaki community.[33][34]
Mohawk languagecode talkers were used duringWorld War IIby theUnited States Armyin the Pacific theater.Levi Oakes, a Mohawk code talker born in Canada, was deployed to protect messages sent by Allied Forces usingKanien'kéha, a Mohawk sub-set language. Oakes died in May 2019; he was the last of the Mohawk code talkers.[35]
TheMuscogee languagewas used as a type two code (informal) during World War II by enlistedSeminoleandCreek peoplein the US Army.[36]Tony Palmer, Leslie Richard,Edmund Harjo, and Thomas MacIntosh from theSeminole Nation of OklahomaandMuscogee (Creek) Nationwere recognized under theCode Talkers Recognition Act of 2008.[37]The last survivor of these code talkers, Edmond Harjo of theSeminole Nation of Oklahoma, died on March 31, 2014, at the age of 96. His biography was recounted at theCongressional Gold Medalceremony honoring Harjo and other code talkers at the US Capitol on November 20, 2013.[38][39][40]
Philip Johnston, a civil engineer for the city of Los Angeles,[41]proposed the use of theNavajo languageto the United States Marine Corps at the beginning of World War II. Johnston, a World War I veteran, was raised on theNavajo reservationas the son of missionaries to the Navajo. He was able to converse in what is called "Trader's Navajo," apidgin language. He was among a few non-Navajo who had enough exposure to it to understand some of its nuances. Many Navajo men enlisted shortly after the attack on Pearl Harbor and eagerly contributed to the war effort.
Because Navajo has a complexgrammar, it is notmutually intelligiblewith even its closest relatives within theNa-Dene familyto provide meaningful information. It was still an unwritten language at the time, and Johnston believed Navajo could satisfy the military requirement for an undecipherable code. Its complex syntax, phonology, and numerous dialects made it unintelligible to anyone without extensive exposure and training. One estimate indicates that fewer than 30 non-Navajo could understand the language during World War II.[42]
In early 1942, Johnston met with the commanding general of the Amphibious Corps, Major GeneralClayton B. Vogel, and his staff. Johnston staged simulated combat conditions, demonstrating that Navajo men could transmit and decode a three-line message in 20 seconds, compared to the 30 minutes it took the machines of the time.[43]The idea of using Navajo speakers as code talkers was accepted; Vogel recommended that the Marines recruit 200 Navajo. However, that recommendation was cut to one platoon to use as a pilot project to develop and test the feasibility of a code. On May 4, 1942, twenty-nine Navajo men were sworn into service atFort Wingate, an old US Army fort converted into aBureau of Indian Affairsboarding school. They were organized as Platoon 382. The first 29 Navajo recruits attended boot camp in May 1942. This first group created the Navajo code atCamp Pendleton.[44]
One of the key features of the Navajo Code Talkers is that they employed a coded version of their language. Other Navajos not trained in the Navajo Code could not decipher the messages being sent.
Platoon 382 was the Marine Corps's first "all-Indian, all-Navajo" Platoon. The members of this platoon would become known asThe First Twenty-Nine. Most were recruited from near the Fort Wingate, NM, area. The youngest was William Dean Yazzie (aka Dean Wilson), who was only 15 when he was recruited. The oldest wasCarl N. Gorman—who with his son, R. C. Gorman, would become an artist of great acclaim and design the Code Talkers' logo—at age 35.
The Navajo code was formally developed and modeled on theJoint Army/Navy Phonetic Alphabetthatuses agreed-upon English words to represent letters. Since it was determined that phonetically spelling out all military terms letter by letter into words while in combat would be too time-consuming, someterms,concepts,tactics, and instruments of modern warfare were given uniquely formal descriptive nomenclatures in Navajo. For example, the word forsharkreferred to a destroyer, whilesilver oak leafindicated the rank of lieutenant colonel.[46]
Acodebookwas developed to teach new initiates the many relevant words and concepts. The text was for classroom purposes only and was never to be taken into the field. The code talkers memorized all these variations and practiced their rapid use under stressful conditions during training. Navajo speakers who had not been trained in the code work would have no idea what the code talkers' messages meant; they would hear only truncated and disjointed strings of individual, unrelated nouns and verbs.[47][48]
The Navajo code talkers were commended for the skill, speed, and accuracy they demonstrated throughout the war. At theBattle of Iwo Jima, Major Howard Connor,5th Marine Divisionsignal officer, had six Navajo code talkers working around the clock during the first two days of the battle. These six sent and received over 800 messages, all without error. Connor later said, "Were it not for the Navajos, the Marines would never have taken Iwo Jima."[44]
After incidents where Navajo code talkers were mistaken for ethnic Japanese and were captured by other American soldiers, several were assigned a personal bodyguard whose principal duty was to protect them from their side. According to Bill Toledo, one of the second groups after the original 29, they had a secret secondary duty: if their charge was at risk of being captured, they were to shoot him to protect the code. Fortunately, none was ever called upon to do so.[49][50]
To ensure consistent use of code terminologies throughout the Pacific theater, representative code talkers of each of the US Marinedivisionsmet in Hawaii to discuss shortcomings in the code, incorporate new terms into the system, and update their codebooks. These representatives, in turn, trained other code talkers who could not attend the meeting. As the war progressed, additional code words were added and incorporated program-wide. In other instances, informal shortcutscode wordswere devised for a particularcampaignand not disseminated beyond the area of operation. Examples of code words include the Navajo word forbuzzard,jeeshóóʼ, which was used forbomber, while the code word used forsubmarine,béésh łóóʼ, meantiron fishin Navajo.[51]The last of the original 29 Navajo code talkers who developed the code,Chester Nez, died on June 4, 2014.[52]
Four of the last nine Navajo code talkers used in the military died in 2019:Alfred K. Newmandied on January 13, 2019, at the age of 94.[53]On May 10, 2019,Fleming Begaye Sr.died at the age of 97.[54]New Mexico State SenatorJohn Pinto, elected in 1977, died in office on May 24, 2019.[55]William Tully Brown died in June 2019 aged 96.[56]Joe Vandever Sr. died at 96 on January 31, 2020.[57]Samuel Sandovaldied on 29 July 2022, at the age of 98.[58][59]John Kinsel Sr.died on 18 October 2024, at the age of 107.[60][61]Only two remaining members are still living as of 2024, Thomas H. Begay and former Navajo chairmanPeter MacDonald.[62]
Some code talkers such as Chester Nez and William Dean Yazzie (aka Dean Wilson) continued to serve in the Marine Corps through the Korean War. Rumors of the deployment of the Navajo code into theKorean Warand after have never been proven. The code remained classified until 1968. The Navajo code is the only spoken military code never to have been deciphered.[46]
In the1973 Arab–Israeli War, Egypt employedNubian-speakingNubian peopleas code talkers.[63][64][65][66][67]
During World War II, American soldiers used their nativeTlingitas a code against Japanese forces. Their actions remained unknown, even after the declassification of code talkers and the publication of the Navajo code talkers. The memory of five deceased Tlingit code talkers was honored by the Alaska legislature in March 2019.[68][69]
A system employing theWelsh languagewas used by British forces during World War II, but not to any great extent. In 1942, the Royal Air Force developed a plan to use Welsh for secret communications, but it was never implemented.[70]Welsh was used more recently in theYugoslav Warsfor non-vital messages.[71]
China usedWenzhounese-speaking people as code talkers during the 1979Sino-Vietnamese War.[72][73]
The Navajo code talkers received no recognition until 1968 when their operation was declassified.[74]In 1982, the code talkers were given a Certificate of Recognition by US PresidentRonald Reagan, who also named August 14, 1982 as Navajo Code Talkers Day.[75][76][77][78]
On December 21, 2000, PresidentBill Clintonsigned Public Law 106–554, 114 Statute 2763, which awarded theCongressional Gold Medalto the original 29 World War II Navajo code talkers andSilver Medalsto each person who qualified as a Navajo code talker (approximately 300). In July 2001, PresidentGeorge W. Bushhonored the code talkers by presenting the medals to four surviving original code talkers (the fifth living original code talker was unable to attend) at a ceremony held in theCapitol Rotundain Washington, DC. Gold medals were presented to the families of the deceased 24 original code talkers.[79][80]
JournalistPatty Talahongvadirected and produced a documentary,The Power of Words: Native Languages as Weapons of War, for theSmithsonian National Museum of the American Indianin 2006, bringing to light the story of Hopi code talkers. In 2011, Arizona established April 23, as an annual recognition day for the Hopi code talkers.[8]TheTexas Medal of Valorwas awarded posthumously to 18 Choctaw code talkers for their World War II service on September 17, 2007, by the Adjutant General of the State of Texas.[81]
The Code Talkers Recognition Act of 2008 (Public Law 110–420)[82]was signed into law by PresidentGeorge W. Bushon November 15, 2008. The Act recognized every Native American code talker who served in the United States military during WWI or WWII (except the already-awarded Navajo) with a Congressional Gold Medal. Approximately 50 tribes were recognized.[83]The act was designed to be distinct for each tribe, with silver duplicates awarded to the individual code talkers or their next-of-kin.[84]As of 2013, 33 tribes have been identified and been honored at a ceremony atEmancipation Hallat the US Capitol Visitor Center. One surviving code talker was present, Edmond Harjo.[85]
On November 27, 2017, three Navajo code talkers, joined by thePresident of the Navajo Nation,Russell Begaye, appeared with PresidentDonald Trumpin theOval Officein an official White House ceremony. They were there to "pay tribute to the contributions of the young Native Americans recruited by the United States military to create top-secret coded messages used to communicate during World War II battles."[86]The executive director of theNational Congress of American Indians,Jacqueline Pata, noted that Native Americans have "a very high level of participation in the military and veterans' service." A statement by a Navajo Nation Council Delegate and comments by Pata and Begaye, among others, objected to Trump's remarks during the event, including his use "once again ... [of] the wordPocahontasin a negative way towards a political adversary Elizabeth Warren who claims 'Native American heritage'."[86][87][88]The National Congress of American Indians objected to Trump's use of the namePocahontas, a historical Native American figure, as a derogatory term.[89]
On March 17, 2025,Axiosreported that numerous articles about Native American Code Talkers were removed from some military websites. According to its reporting,Axiosidentified at least 10 articles which had disappeared from the U.S. Army and Department of Defense websites. Pentagon Press Secretary John Ullyot is quoted in response: "As Secretary [Pete] Hegseth has said, DEI is dead at the Defense Department. ... We are pleased by the rapid compliance across the Department with the directive removing DEI content from all platforms."[90][91]
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ATM SafetyPIN softwareis asoftware applicationthat allows users ofautomated teller machines(ATMs) to alert law enforcement of a forced cash withdrawal (such as in arobbery) by entering theirpersonal identification number(PIN) in reverse order.[1]The system was patented byIllinoislawyer Joseph Zingher (U.S. patent 5,731,575).
SafetyPIN is not currently used in ATM systems, despite widely circulated rumors originating from achain lettere-mail. This is mainly due to issues regardingpalindromePINs being incompatible with the system and potentialsecurity vulnerabilitiesthat could arise if implemented.
The concept of a backup emergency PIN system, orduress code, for ATM systems has been around since at least July 30, 1986, whenRepresentativeMario Biaggi, a former police officer, proposed it in the U.S.Congressional Record, pp. 18232 et seq. Biaggi then proposed House Resolution 785 in 1987 which would have had theFBItrack the problem ofexpress kidnappingsand evaluate the idea of an emergency PIN system. HR785 died in committee without debate.
Zingher has not been successful in marketing licenses for his patent.[2]Police in New York, New Jersey, Ohio, Illinois, and Kansas have supported the concept.[3][4][5][6]Police support prompted the Illinois legislature to pass a law making it mandatory on all ATMs in Illinois. The law was changed shortly after it was passed by a "follow-on" bill that changed the meaning to the exact opposite of what they were seeking.[7][8][9][10]
In 2006, an e-mailchain letterhoaxcirculated that claimed a reverse PIN duress code system is in place universally.[11]American Bankerreported on January 2, 2007, that no PIN-reversal duress code is used on any ATM as of that date.
In September 2013 the hoax was still circulating in Australia with the text:
If you should ever be forced by a robber to withdraw money from an ATM, you can notify the police by entering your PIN in reverse. For example if your PIN is 1234 then you would put in 4321. The ATM recognizes that your PIN is backwards from the ATM card you placed in the machine. The machine will still give you the money you requested, but unknown to the robber, the police will be immediately dispatched to help you. This information was recently broadcast on TV and it states that it is seldom used because people don't know it exists. Please pass this along to everyone possible. Australian Federal Police. AFP Web site:https://www.afp.gov.au
The same kind of e-mailchain letterhoaxis still circulated onTumblrandFacebook, as well as inIndiaand other parts of the world.
Were the system implemented,palindromicPINs such as 5555 or 2112 would then be unavailable so that false alarms would not occur. Moreover, PINs that are semi-reversible such as 5255 or 1241, where the first and last numbers are the same, would be something to avoid as well so that accidental alarms would not be triggered by mistakenly switching the middle numbers.
Diebold, a manufacturer of ATMs, states on their website that no such emergency alerting system is currently in use. They cite an article in the St. Louis Post-Dispatch which claims bankers oppose the reverse-PIN system out of concerns that "ATM users might hesitate or fumble while trying to enter their PINs backwards under duress, possibly increasing the chances of violence." Diebold further states that they would be willing to support such technology if their customers (presumably banks) request it.[11]
A bill making the reverse emergency PIN system mandatory on all ATMs in the state of Illinois was proposed on February 10, 2009. Subsection (i) is the new bill.[12]
i) A terminal operated in this State must be designed and programmed so that when a consumer enters his or her personal identification number in reverse order, the terminal automatically sends an alarm to the local law enforcement agency having jurisdiction over the terminal location. The Commissioner shall promulgate rules necessary for the implementation of this subsection (i).
In 2009, Los Angeles City Councilman Greig Smith announced his intention to make the ReversePIN system mandatory on all ATMs in the city.[13][14]
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https://en.wikipedia.org/wiki/ATM_SafetyPIN_software
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Atransaction authentication number(TAN) is used by someonline bankingservices as a form ofsingle useone-time passwords(OTPs) to authorizefinancial transactions. TANs are a second layer of security above and beyond the traditional single-passwordauthentication.
TANs provide additional security because they act as a form oftwo-factor authentication(2FA). If the physical document or token containing the TANs is stolen, it will be useless without the password. Conversely, if the login data are obtained, no transactions can be performed without a valid TAN.
TANs often function as follows:
However, as any TAN can be used for any transaction, TANs are still prone tophishing attackswhere the victim is tricked into providing both password/PIN and one or several TANs. Further, they provide no protection againstman-in-the-middle attacks, where an attacker intercepts the transmission of the TAN, and uses it for a forged transaction, such as when the client system becomes compromised by some form ofmalwarethat enables amalicious user. Although the remaining TANs are uncompromised and can be used safely, users are generally advised to take further action, as soon as possible.
Indexed TANs reduce the risk of phishing. To authorize a transaction, the user is not asked to use an arbitrary TAN from the list but to enter a specific TAN as identified by a sequence number (index). As the index is randomly chosen by the bank, an arbitrary TAN acquired by an attacker is usually worthless.
However, iTANs are still susceptible toman-in-the-middle attacks, including phishing attacks where the attacker tricks the user into logging into a forged copy of the bank's website andman-in-the-browser attacks[2]which allow the attacker to secretly swap the transaction details in the background of the PC as well as to conceal the actual transactions carried out by the attacker in the online account overview.[3]
Therefore, in 2012 theEuropean Union Agency for Network and Information Securityadvised all banks to consider the PC systems of their users being infected bymalwareby default and use security processes where the user can cross-check the transaction data against manipulations like for example (provided the security of the mobile phone holds up)mTANor smartcard readers with their own screen including the transaction data into the TAN generation process while displaying it beforehand to the user (chipTAN).[4]
Prior to entering the iTAN, the user is presented aCAPTCHA, which in the background also shows the transaction data and data deemed unknown to a potential attacker, such as the user's birthdate. This is intended to make it hard (but not impossible) for an attacker to forge the CAPTCHA.
This variant of the iTAN is method used by some German banks adds aCAPTCHAto reduce the risk of man-in-the-middle attacks.[5]Some Chinese banks have also deployed a TAN method similar to iTANplus. A recent study shows that these CAPTCHA-based TAN schemes are not secure against more advanced automated attacks.[6]
mTANs are used by banks in Austria, Bulgaria, Czech Republic, Germany, Hungary, Malaysia, the Netherlands, Poland, Russia, Singapore, South Africa, Spain, Switzerland and some in New Zealand, Australia, UK, and Ukraine. When the user initiates a transaction, a TAN is generated by the bank and sent to the user's mobile phone bySMS. The SMS may also include transaction data, allowing the user to verify that the transaction has not been modified in transmission to the bank.
However, the security of this scheme depends on the security of the mobile phone system. In South Africa, where SMS-delivered TAN codes are common, a new attack has appeared: SIM Swap Fraud. A common attack vector is for the attacker toimpersonatethe victim, and obtain a replacementSIM cardfor the victim's phone from themobile network operator. The victim's user name and password are obtained by other means (such askeyloggingorphishing). In-between obtaining the cloned/replacement SIM and the victim noticing their phone no longer works, the attacker can transfer/extract the victim's funds from their accounts.[7]In 2016 astudy was conducted on SIM Swap Fraudby asocial engineer, revealing weaknesses in issuing porting numbers.
In 2014, a weakness in theSignalling System No. 7used for SMS transmission was published, which allows interception of messages. It was demonstrated by Tobias Engel during the 31stChaos Communication Congress.[8]At the beginning of 2017, this weakness was used successfully in Germany to intercept SMS and fraudulently redirect fund transfers.[9]
Also the rise ofsmartphonesled to malware attacks trying to simultaneously infect the PC and the mobile phone as well to break the mTAN scheme.[10]
pushTAN is anapp-based TAN scheme by German Sparkassen banking group reducing some of the shortcomings of themTANscheme. It eliminates the cost of SMS messages and is not susceptible to SIM card fraud, since the messages are sent via a special text-messaging application to the user's smartphone using an encrypted Internet connection. Just like mTAN, the scheme allows the user to cross-check the transaction details against hidden manipulations carried out byTrojanson the user's PC by including the actual transaction details the bank received in the pushTAN message. Although analogous to using mTAN with a smartphone, there is the risk of a parallel malware infection of PC and smartphone. To reduce this risk the pushTAN app ceases to function if the mobile device isrootedor jailbroken.[11]In late 2014 the Deutsche Kreditbank (DKB) also adopted the pushTAN scheme.[12]
The risk of compromising the whole TAN list can be reduced by usingsecurity tokensthat generate TANs on-the-fly, based on a secret known by the bank and stored in the token or a smartcard inserted into the token.
However, the TAN generated is not tied to the details of a specific transaction. Because the TAN is valid for any transaction submitted with it, it does not protect againstphishingattacks where the TAN is directly used by the attacker, or againstman-in-the-middle attacks.
ChipTAN is a TAN scheme used by many German and Austrian banks.[13][14][15]It is known as ChipTAN or Sm@rt-TAN[16]in Germany and as CardTAN in Austria, whereas cardTAN is a technically independent standard.[17]
A ChipTAN generator is not tied to a particular account; instead, the user must insert theirbank cardduring use. The TAN generated is specific to the bank card as well as to the current transaction details. There are two variants: In the older variant, the transaction details (at least amount and account number) must be entered manually.In the modern variant, the user enters the transaction online, then the TAN generator reads the transaction details via a flickeringbarcodeon the computer screen (usingphotodetectors). It then shows the transaction details on its own screen to the user for confirmation before generating the TAN.
As it is independent hardware, coupled only by a simple communication channel, the TAN generator is not susceptible to attack from the user's computer. Even if the computer is subverted by aTrojan, or if aman-in-the-middle attackoccurs, the TAN generated is only valid for the transaction confirmed by the user on the screen of the TAN generator, therefore modifying a transaction retroactively would cause the TAN to be invalid.
An additional advantage of this scheme is that because the TAN generator is generic, requiring a card to be inserted, it can be used with multiple accounts across different banks, and losing the generator is not a security risk because the security-critical data is stored on the bank card.
While it offers protection from technical manipulation, the ChipTAN scheme is still vulnerable tosocial engineering. Attackers have tried to persuade the users themselves to authorize a transfer under a pretext, for example by claiming that the bank required a "test transfer" or that a company had falsely transferred money to the user's account and they should "send it back".[2][18]Users should therefore never confirm bank transfers they have not initiated themselves.
ChipTAN is also used to secure batch transfers (Sammelüberweisungen). However, this method offers significantly less security than the one for individual transfers. In case of a batch transfer the TAN generator will only show the number and total amount of all transfers combined – thus for batch transfers there is little protection from manipulation by a Trojan.[19]This vulnerability was reported by RedTeam Pentesting in November 2009.[20]In response, as a mitigation, some banks changed their batch transfer handling so that batch transfers containing only a single record are treated as individual transfers.
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https://en.wikipedia.org/wiki/Transaction_authentication_number
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TheAztec Codeis amatrix codeinvented by Andrew Longacre, Jr. and Robert Hussey in 1995.[1]The code was published byAIM, Inc.in 1997. Although the Aztec Code waspatented, that patent was officially madepublic domain.[2]The Aztec Code is also published as ISO/IEC 24778:2024 standard. Named after the resemblance of the central finder pattern to anAztec pyramid, Aztec Code has the potential to use less space than other matrix barcodes because it does not require a surrounding blank "quiet zone".
The symbol is built on a square grid with a bull's-eye pattern at its centre for locating the code. Data isencodedin concentric square rings around the bull's-eye pattern. The central bull's-eye is 9×9 or 13×13pixels, and one row of pixels around that encodes basic coding parameters, producing a "core" of 11×11 or 15×15 squares. Data is added in "layers", each one containing two rings of pixels, giving total sizes of 15×15, 19×19, 23×23, etc.
The corners of the core include orientation marks, allowing the code to be read if rotated or reflected. Decoding begins at the corner with three black pixels, and proceeds clockwise to the corners with two, one, and zero black pixels. The variable pixels in the central core encode the size, so it is not necessary to mark the boundary of the code with a blank "quiet zone", although some barcode readers require one.
The compact Aztec code core may be surrounded by 1 to 4 layers, producing symbols from 15×15 (room for 13 digits or 12 letters) through 27×27. There is additionally a special 11×11 "rune" that encodes one byte of information. The full core supports up to 32 layers, 151×151 pixels, which can encode 3832 digits, 3067 letters, or 1914 bytes of data.
Whatever part of the symbol is not used for the basic data is used forReed–Solomon error correction, and the split is completely configurable, between limits of 1 data word, and 3 check words. The recommended number of check words is 23% of symbol capacity plus 3 codewords.[3]
Aztec Code is supposed to produce readable codes with various printer technologies. It is also well suited for displays of cell phones and other mobile devices.
The encoding process consists of the following steps:
All conversion between bits strings and other forms is performed according to thebig-endian(most significant bit first) convention.
All 8-bit values can be encoded, plus two escape codes:
By default, codes 0–127 are interpreted according to ANSI X3.4 (ASCII), and 128–255 are interpreted according toISO/IEC 8859-1: Latin Alphabet No. 1. This corresponds to ECI 000003.
Bytes are translated into 4- and 5-bit codes, based on a current decoding mode, with shift and latch codes for changing modes. Byte values not available this way may be encoded using a general "binary shift" code, which is followed by a length and a number of 8-bit codes.
For changing modes, ashiftaffects only the interpretation of the single following code, while alatchaffects all following codes. Most modes use 5-bit codes, but Digit mode uses 4-bit codes.
B/S (binary shift) is followed by a 5-bit length. If non-zero, this indicates that 1–31 8-bit bytes follow. If zero, 11 additional length bits encode the number of following bytes less 31. (Note that for 32–62 bytes, two 5-bit byte shift sequences are more compact than one 11-bit.) At the end of the binary sequence, the previous mode is resumed.
FLG(n) is followed by a 3-bitnvalue.n=0 encodes FNC1.n=1–6 is followed by 1–6 digits (in digit mode) which are zero-padded to make a 6-bit ECI identifier.n=7 is reserved and currently illegal.
The mode message encodes the number of layers (Llayers encoded as the integerL−1), and the number of data codewords (Dcodewords, encoded as the integerD−1) in the message. All remaining codewords are used as check codewords.
For compact Aztec codes, the number of layers is encoded as a 2-bit value, and the number of data codewords as a 6-bit value, resulting in an 8-bit mode word. For full Aztec codes, the number of layers is encoded in 5 bits, and the number of data codewords is encoded in 11 bits, making a 16-bit mode word.
The mode word is broken into two or four 4-bit codewords inGF(16), and 5 or 6 Reed–Solomon check words are appended, making a 28- or 40-bit mode message, which is wrapped in a 1-pixel layer around the core. Thus a (15,10) or (15,9) Reed-Solomon code (shortened to (7,2) or (10,4) respectively), over GF(16) is used.
Because an L+1-layer compact Aztec code can hold more data than an L-layer full code, full codes with less than 4 layers are rarely used.
Most importantly, the number of layers determines the size of the Reed–Solomon codewords used. This varies from 6 to 12 bits:
The codeword sizebis the smallest even number which ensures that the total number of codewords in the symbol is less than the limit of 2b−1 which can be corrected by a Reed–Solomon code.
As mentioned above, it is recommended that at least 23% of the available codewords, plus 3, are reserved for correction, and a symbol size is chosen such that the message will fit into the available space.
The data bits are broken into codewords, with the first bit corresponding to the most significant coefficient. While doing this, code words of all-zero and all-ones are avoided bybit stuffing: if the firstb−1 bits of a code word have the same value, an extra bit with the complementary value is inserted into the data stream. This insertion takes place whether or not the last bit of the code word would have had the same value or not.
Also, note that this only applies to strings ofb−1 bitsat the beginning of a code word. Longer strings of identical bits are permitted as long as they straddle a code word boundary.
When decoding, a code word of all zero or all one may be assumed to be anerasure, and corrected more efficiently than a general error.
This process makes the message longer, and the final number of data codewords recorded in the mode message is not known until it is complete. In rare cases, it may be necessary to jump to the next-largest symbol and begin the process all over again to maintain the minimum fraction of check words.
After bit stuffing, the data string is padded to the next codeword boundary by appending 1 bit. If this would result in a code word of all ones, the last bit is changed to zero (and will be ignored by the decoder as a bit-stuffing bit). On decoding, the padding bits may be decoded as shift and latch codes, but that will not affect the message content. The reader must accept and ignore a partial code at the end of the message, as long as it is all-ones.
Additionally, if the total number of data bits available in the symbol is not a multiple of the codeword size, the data string is prefixed with an appropriate number of 0 bits to occupy the extra space. These bits are not included in the check word computation.
Both the mode word, and the data, must have check words appended to fill out the available space. This is computed by appendingKcheck words such that the entire message is a multiple of the Reed–Solomon polynomial (x−2)(x−4)...(x−2K).
Note that check words arenotsubject to bit stuffing, and may be all-zero or all-one. Thus, it is not possible to detect the erasure of a check word.
A full Aztec code symbol has, in addition to the core, a "reference grid" of alternating black and white pixels occupying every 16th row and column. A compact Aztec code does not contain this grid. These known pixels allow a reader to maintain alignment with the pixel grid over large symbols. For up to 4 layers (31×31 pixels), this consists only of single lines extending outward from the core, continuing the alternating pattern. Inside the 5th layer, however, additional rows and columns of alternating pixels are inserted ±16 pixels from the center, so the 5th layer is located ±17 and ±18 pixels from the center, and a 5-layer symbol is 37×37 pixels.
Likewise, additional reference grid rows and columns are inserted ±32 pixels from the center, making a 12-layer symbol 67×67 pixels. In this case, the 12th layer occupies rings ±31 and ±33 pixels from the center. The pattern continues indefinitely outward, with 15-pixel blocks of data separated by rows and columns of the reference grid.
One way to construct the symbol is to delete the reference grid entirely and begin with a 14×14-pixel core centered on a 2×2 pixel-white square. Then break it into 15×15 pixel blocks and insert the reference grid between them.
The mode message begins at the top-left corner of the core and wraps around it clockwise in a 1-bit thick layer. It begins with the most significant bit of the number of layers and ends with the check words. For a compact Aztec code, it is broken into four 7-bit pieces to leave room for the orientation marks. For a full Aztec code, it is broken into four 10-bit pieces, and those pieces are each divided in half by the reference grid.
In some cases, the total capacity of the matrix does not divide evenly by full code words. In such cases, the main message is padded with 0 bits in the beginning. These bits are not included in the check word calculation and should be skipped during decoding. The total matrix capacity for a full symbol can be calculated as (112+16*L)*L for a full Aztec code and (88+16*L)*L for a compact Aztec code, where L is the symbol size in layers.[4]As an example, the total matrix capacity of a compact Aztec code with 1 layer is 104 bits. Since code words are six bits, this gives 17 code words and two extra bits. Two zero bits are prepended to the message as padding and must be skipped during decoding.
The padded main message begins at the outer top-left of the entire symbol and spirals around itcounterclockwisein a 2-bit thick layer, ending directly above the top-left corner of the core. This places the bit-stuffed data words, for which erasures can be detected, in the outermost layers of the symbol, which are most prone to erasures. The check words are stored closer to the core. The last check word ends just above the top left corner of the bull's eye.
With the core in its standard orientation, the first bit of the first data word is placed in the upper-left corner, with additional bits placed in a 2-bit-wide column left-to-right and top-to-bottom. This continues until 2 rows from the bottom of the symbol when the pattern rotates 90 degrees counterclockwise and continues in a 2-bit high row, bottom-to-top and left-to-right. After 4 equal-sized quarter layers, the spiral continues with the top-left corner of the next-inner layer, finally ending one pixel above the top-left corner of the core.
Finally, 1 bit are printed as black squares, and 0 bits are printed as white squares.
Aztec codes are widely used for transport ticketing.
The Aztec Code has been selected by the airline industry (IATA'sBCBPstandard) for electronic boarding passes. Several airlines send Aztec Codes to passengers' mobile phones to act as boarding passes. These are often integrated with apps on passengers' phones, includingApple Wallet.
Aztec codes are also used in rail, including byTehran Metro, BritishNational Rail,[5]Eurostar,Deutsche Bahn,TCDD Taşımacılık,DSB,SJ,České dráhy,Slovak Railways,Slovenian Railways,Croatian Railways,Trenitalia,Nederlandse Spoorwegen,Pasažieru vilciens,PKP Intercity,VR Group,Via Rail,Swiss Federal Railways,SNCBandSNCFfor tickets sold online and printed out by customers or displayed on mobile phone screens. The Aztec code is scanned by a handheld scanner by on-train staff or at the turnstile to validate the ticket.
Car registration documents inPolandbear a summary, compressed by NRV2E algorithm, encoded as Aztec Code. Works are underway to enable car insurance companies to automatically fill in the relevant information based on digital photographs of the document as the first step of closing a new insurance contract.
Federal Tax ServiceinRussiaencodes payment information in tax notices as Aztec Code.
Manybillsin Canada are now using this technology as well, includingEastLink,Shaw Cable, andBell Aliant.
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https://en.wikipedia.org/wiki/Aztec_Code
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AData Matrixis atwo-dimensional codeconsisting of black and white "cells" or dots arranged in either asquareorrectangularpattern, also known as amatrix. The information to be encoded can be text or numeric data. Usual data size is from a few bytes up to 1556bytes. The length of the encoded data depends on the number of cells in the matrix.Error correction codesare often used to increase reliability: even if one or more cells are damaged so it is unreadable, the message can still be read. A Data Matrix symbol can store up to 2,335alphanumericcharacters.
Data Matrix symbols are rectangular, usually square in shape and composed of square "cells" which representbits. Depending on the coding used, a "light" cell represents a 0 and a "dark" cell is a 1, or vice versa. Every Data Matrix is composed of two solid adjacent borders in an "L" shape (called the "finder pattern") and two other borders consisting of alternating dark and light "cells" or modules (called the "timing pattern"). Within these borders are rows and columns of cells encoding information. The finder pattern is used to locate and orient the symbol while the timing pattern provides a count of the number of rows and columns in the symbol. As more data is encoded in the symbol, the number of cells (rows and columns) increases. Each code is unique. Symbol sizes vary from 10×10 to 144×144 in the new version ECC 200, and from 9×9 to 49×49 in the old version ECC 000 – 140.
The most popular application for Data Matrix is marking small items, due to the code's ability to encode fifty characters in a symbol that is readable at 2 or 3 mm2(0.003 or 0.005 sq in) and the fact that the code can be read with only a 20% contrast ratio.[1]A Data Matrix is scalable; commercial applications exist with images as small as 300 micrometres (0.012 in) (laser etched on a 600-micrometre (0.024 in) silicon device) and as large as a 1 metre (3 ft) square (painted on the roof of aboxcar). Fidelity of the marking and reading systems are the only limitation.
The USElectronic Industries Alliance(EIA) recommends using Data Matrix for labeling small electronic components.[2]
Data Matrix codes are becoming common on printed media such as labels and letters. The code can be read quickly by abarcode readerwhich allows the media to be tracked, for example when a parcel has been dispatched to the recipient.
For industrial engineering purposes, Data Matrix codes can be marked directly onto components, ensuring that only the intended component is identified with the data-matrix-encoded data. The codes can be marked onto components with various methods, but within the aerospace industry these are commonly industrial ink-jet, dot-peen marking, laser marking, and electrolytic chemical etching (ECE). These methods give a permanent mark which can last up to the lifetime of the component.
Data Matrix codes are usually verified using specialist camera equipment and software.[further explanation needed]This verification ensures the code conforms to the relevant standards, and ensures readability for the lifetime of the component. After component enters service, the Data Matrix code can then be read by a reader camera, which decodes the Data Matrix data which can then be used for a number of purposes, such as movement tracking or inventory stock checks.
Data Matrix codes, along with other open-source codes such as 1D barcodes can also be read with mobile phones by downloading code specific mobile applications. Although many mobile devices are able to read 2D codes including Data Matrix Code,[3]few extend the decoding to enable mobile access and interaction, whereupon the codes can be used securely and across media; for example, in track and trace, anti-counterfeit, e.govt, and banking solutions.
Data Matrix codes are used in thefood industryinautocodingsystems to prevent food products being packaged and dated incorrectly. Codes are maintained internally on a food manufacturers database and associated with each unique product, e.g. ingredient variations. For each product run the unique code is supplied to the printer. Label artwork is required to allow the 2D Data Matrix to be positioned for optimal scanning. For black on white codes testing isn't required unless print quality is an issue, but all color variations need to be tested before production to ensure they are readable.[citation needed]
In May 2006 a German computer programmer, Bernd Hopfengärtner, created a large Data Matrix in a wheat field (in a fashion similar tocrop circles). The message read "Hello, World!".[4]
Data Matrix symbols are made up of modules arranged within a perimeter finder and timing pattern. It can encode up to 3,116 characters from the entireASCIIcharacter set (with extensions). The symbol consists of data regions which contain modules set out in a regular array. Large symbols contain several regions. Each data region is delimited by a finder pattern, and this is surrounded on all four sides by a quiet zone border (margin). (Note: The modules may be round or square- no specific shape is defined in the standard. For example, dot-peened cells are generally round.)
ECC 200, the newer version of Data Matrix, usesReed–Solomoncodes for error and erasure recovery. ECC 200 allows the routine reconstruction of the entire encoded data string when the symbol has sustained 30% damage, assuming the matrix can still be accurately located. Data Matrix has an error rate of less than 1 in 10 million characters scanned.[5]
Symbols have an even number of rows and an even number of columns. Most of the symbols are square with sizes from 10 × 10 to 144 × 144. Some symbols however are rectangular with sizes from 8×18 to 16×48 (even values only). All symbols using the ECC 200 error correction can be recognized by the upper-right corner module being the same as the background color. (binary 0).
Additional capabilities that differentiate ECC 200 symbols from the earlier standards include:
[6]
Older versions of Data Matrix include ECC 000, ECC 050, ECC 080, ECC 100, ECC 140. Instead of usingReed–Solomoncodes like ECC 200, ECC 000–140 use a convolution-based error correction. Each varies in the amount of error correction it offers, with ECC 000 offering none, and ECC 140 offering the greatest. For error detection at decode time, even in the case of ECC 000, each of these versions also encode acyclic redundancy check(CRC) on the bit pattern. As an added measure, the placement of each bit in the code is determined by bit-placement tables included in the specification. These older versions always have an odd number of modules, and can be made in sizes ranging from 9 × 9 to 49 × 49. All symbols utilizing the ECC 000 through 140 error correction can be recognized by the upper-right corner module being the inverse of the background color. (binary 1).
According to ISO/IEC 16022, "ECC 000–140 should only be used in closed applications where a single party controls both the production and reading of the symbols and is responsible for overall system performance."
Data Matrix was invented byInternational Data Matrix, Inc.(ID Matrix) which was merged into RVSI/Acuity CiMatrix, who were acquired bySiemensAG in October 2005 and Microscan Systems in September 2008. Data Matrix is covered today by severalISO/IECstandards and is in the public domain for many applications, which means it can be used free of any licensing or royalties.
Data Matrix codes useReed–Solomon error correctionover thefinite fieldF256{\displaystyle \mathbb {F} _{256}}(orGF(28)), the elements of which are encoded asbytes of 8 bits; the byteb7b6b5b4b3b2b1b0{\displaystyle b_{7}b_{6}b_{5}b_{4}b_{3}b_{2}b_{1}b_{0}}with a standard numerical value∑i=07bi2i{\displaystyle \textstyle \sum _{i=0}^{7}b_{i}2^{i}}encodes the field element∑i=07biαi{\displaystyle \textstyle \sum _{i=0}^{7}b_{i}\alpha ^{i}}whereα∈F256{\displaystyle \alpha \in \mathbb {F} _{256}}is taken to be a primitive element satisfyingα8+α5+α3+α2+1=0{\displaystyle \alpha ^{8}+\alpha ^{5}+\alpha ^{3}+\alpha ^{2}+1=0}. The primitive polynomial isx8+x5+x3+x2+1{\displaystyle x^{8}+x^{5}+x^{3}+x^{2}+1}, corresponding to the polynomial number 301, with initial root = 1 to obtain generator polynomials. The Reed–Solomon code uses different generator polynomials overF256{\displaystyle \mathbb {F} _{256}}, depending on how many error correction bytes the code adds. The number of bytes added is equal to the degree of the generator polynomial.
For example, in the 10 × 10 symbol, there are 3 data bytes and 5 error correction bytes. The generator polynomial is obtained as:g(x)=(x+α)(x+α2)(x+α3)(x+α4)(x+α5){\displaystyle g(x)=(x+\alpha )(x+\alpha ^{2})(x+\alpha ^{3})(x+\alpha ^{4})(x+\alpha ^{5})},
which gives:g(x)=x5+α235x4+α207x3+α210x2+α244x+α15{\displaystyle g(x)=x^{5}+\alpha ^{235}x^{4}+\alpha ^{207}x^{3}+\alpha ^{210}x^{2}+\alpha ^{244}x+\alpha ^{15}},
or with decimal coefficients:g(x)=x5+62x4+111x3+15x2+48x+228{\displaystyle g(x)=x^{5}+62x^{4}+111x^{3}+15x^{2}+48x+228}.
The encoding process is described in theISO/IECstandard 16022:2006.[7]Open-source software for encoding and decoding the ECC-200 variant of Data Matrix has been published.[8][9]
The diagrams below illustrate the placement of the message data within a Data Matrix symbol. The message is "Wikipedia", and it is arranged in a somewhat complicated diagonal pattern starting near the upper-left corner. Some characters are split in two pieces, such as the initial W, and the third 'i' is in "corner pattern 2" rather than the usual L-shaped arrangement. Also shown are the end-of-message code (marked End), the padding (P) and error correction (E) bytes, and four modules of unused space (X).
The symbol is of size 16×16 (14×14 data area), with 12 data bytes (including 'End' and padding) and 12 error correction bytes. A (255,243,6) Reed Solomon code shortened to (24,12,6) is used. It can correct up to 6 byte errors or erasures.
To obtain the error correction bytes, the following procedure may be carried out:
The generator polynomial specified for the (24,12,6) code, is:g(x)=x12+242x11+100x10+178x9+97x8+213x7+142x6+42x5+61x4+91x3+158x2+153x+41{\displaystyle g(x)=x^{12}+242x^{11}+100x^{10}+178x^{9}+97x^{8}+213x^{7}+142x^{6}+42x^{5}+61x^{4}+91x^{3}+158x^{2}+153x+41},
which may also be written in the form of a matrix of decimal coefficients:
The 12-byte long message "Wikipedia" including 'End', P1 and P2, in decimal coefficients (see the diagrams below for the computation method using ASCII values), is:
Using the procedure forReed-Solomon systematic encoding, the 12 error correction bytes obtained (E1 through E12 in decimal) in the form of the remainder after polynomial division are:
These error correction bytes are then appended to the original message. The resulting coded message has 24 bytes, and is in the form:
or in decimal coefficients:
and in hexadecimal coefficients:
Multiple encoding modes are used to store different kinds of messages. The default mode stores oneASCIIcharacter per 8-bit codeword. Control codes are provided to switch between modes, as shown below.
The C40, Text andX12modes are potentially more compact for storing text messages. They are similar toDEC Radix-50, using character codes in the range 0–39, and three of these codes are combined to make a number up to 403=64000, which is packed into two bytes (maximum value 65536) as follows:
The resulting value of B1 is in the range 0–250. The special value 254 is used to return to ASCII encoding mode.
Character code interpretations are shown in the table below. The C40 and Text modes have four separate sets. Set 0 is the default, and contains codes that temporarily select a different set for the next character. The only difference is that they reverse upper-and lower-case letters. C40 is primarily upper-case, with lower-case letters in set 3; Text is the other way around. Set 1, containing ASCII control codes, and set 2, containing punctuation symbols are identical in C40 and Text mode.
EDIFACTmode uses six bits per character, with four characters packed into three bytes. It can store digits, upper-case letters, and many punctuation marks, but has no support for lower-case letters.
Base 256 mode data starts with a length indicator, followed by a number of data bytes. A length of 1 to 249 is encoded as a single byte,
and longer lengths are stored as two bytes.
It is desirable to avoid long strings of zeros in the coded message, because they become large blank areas in the Data Matrix symbol, which may
cause a scanner to lose synchronization. (The default ASCII encoding does not use zero for this reason.) In order to make that less likely, the
length and data bytes are obscured by adding a pseudorandom value R(n), where n is the position in the byte stream.
Prior to the expiration of US patent 5,612,524[10]in November 2007, intellectual property companyAcacia Technologiesclaimed that Data Matrix was partially covered by its contents. As the patent owner, Acacia allegedly contacted Data Matrix users demanding license fees related to the patent.
Cognex Corporation, a large manufacturer of 2D barcode devices, filed adeclaratory judgmentcomplaint on 13 March 2006 after receiving information that Acacia had contacted its customers demanding licensing fees. On 19 May 2008 Judge Joan N. Ericksen of the U.S. District Court in Minnesota ruled in favor of Cognex.[11]The ruling held that the '524 patent, which claimed to cover a system for capturing and reading 2D symbology codes, is both invalid and unenforceable due toinequitable conductby the defendants during the procurement of the patent.
While the ruling was delivered after the patent expired, it precluded claims for infringement based on use of Data Matrix prior to November 2007.
A German patent application DE 4107020 was filed in 1991, and published in 1992. This patent is not cited in the above US patent applications and might invalidate them.[citation needed]
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https://en.wikipedia.org/wiki/Data_Matrix
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High Capacity Color Barcode(HCCB) is a technology developed byMicrosoftfor encoding data in a2D "barcode"using clusters of colored triangles instead of the square pixels conventionally associated with 2D barcodes orQR codes.[1]Data density is increased by using a palette of 4 or 8 colors for the triangles, although HCCB also permits the use of black and white when necessary. It has been licensed by the ISAN International Agency for use in itsInternational Standard Audiovisual Numberstandard,[2]and serves as the basis for the Microsoft Tagmobile taggingapplication.
The technology was created byGavin Jancke, an engineering director atMicrosoft Research. Quoted by BBC News in 2007, he said that HCCB was not intended to replace conventionalbarcodes. "'It's more of a 'partner' barcode", he said. "TheUPCbarcodes will always be there. Ours is more of a niche barcode where you want to put a lot of information in a small space."[3]
HCCB uses a grid of colored triangles to encode data. Depending on the target use, the grid size (total number of symbols), symbol density (the printed size of the triangles), and symbol count (number of colors used) can be varied. HCCB can use an eight-, four-, or two-color (black-and-white) palette. Microsoft claims that laboratory tests using standard off-the-shelf printers and scanners have yielded readable eight-color HCCBs equivalent to approximately 3,500 characters per square inch.[1][3]
Microsoft Tagis a discontinued but still available implementation of High Capacity Color Barcode (HCCB) using 4 colors in a 5 x 10 grid. Additionally, the code works in monochrome.[4]The print size can be varied to allow reasonable reading by a mobile camera phone; for example, a Tag on a real estate sign might be printed large enough to be read from a car driving by, whereas a Tag in a magazine could be smaller because the reader would likely be nearer.
A Microsoft Tag is essentially a machine readableweb link, analogous to aURL shorteninglink: when read, the Tag application sends the HCCB data to a Microsoft server, which then returns the publisher's intended URL. The Tag reader then directs the user'smobile browserto the appropriate website. Because of this redirection, Microsoft is also able to track users and provide Taganalyticsto publishers.
When the platform was released, creation of tags for both commercial and noncommercial use was free as were the associated analytics.[5]In 2013, the process for creating new accounts was transferred to Scanbuy, which said that "A free plan will also be offered from ScanLife with the same basic features", although additional features may be available at extra cost.[6]
Users can download the free Microsoft Tag reader application to their Internet-capable mobile device with camera, launch the reader and read a tag using their phone’s camera. Depending on the scenario, this triggers the intended content to be displayed. SomeGPS-equipped phones can, at the user's option, send coordinate data along with the HCCB data, allowing location-specific information to be returned (e.g. for a restaurant advertisement, a navigational map to the nearest location could be shown).[7]
The Microsoft Tag application gives people the ability to use a mobile phone's on-board camera to take a picture of a tag, and be directed to information in any form, such as text,vCard,URL, Online Photos, Online Video or contact details for the publisher.
Two-dimensional tags can be used to transform traditional marketing media (for example, print advertising, billboards, packaging and merchandising in stores or on LCDs) into gateways for accessing information online. Tags can be applied as gateways from any type of media to an internet site or online media.
The Microsoft Tag reader application is a free download for an Internet-capable mobile device with a camera. The Microsoft Tag reader is compatible with Internet-capable mobile devices, including many based on theWindows Phone 7,Windows Mobile,BlackBerry,Java,Android,Symbian S60,iPhoneandJava MEplatforms.[8]
On August 19, 2013 Microsoft sent out an email notice that the Microsoft Tag service will be terminated in two years on August 19, 2015. Scanbuy, a company founded in 2000 by Olivier Attia, has been selected to support Microsoft Tag technology on the ScanLife platform beginning September 18, 2013.
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https://en.wikipedia.org/wiki/High_Capacity_Color_Barcode
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JAB Code (Just Another Barcode)is a colour 2D matrix symbology made of colour squares arranged in either square or rectangle grids. It was developed byFraunhofer Institute for Secure Information Technology SIT[de].[1]
The code contains one primary symbol and optionally multiple secondary symbols. The primary symbol contains four finder patterns located at the corners of the symbol.[2]
The code uses either four or eight colours.[3]The four basic colours (cyan, magenta, yellow, and black) are the four primary colours of the subtractiveCMYK colour model, which is the most widely used system in the industry for colour printing on a white base such as paper. The other four colours (blue, red, green, and white) are secondary colours of the CMYK model and each originates as an equal mixture of a pair of basic colours.
The barcode is not subject to licensing and was submitted to ISO/IEC standardization as ISO/IEC 23634 expected to be approved at the beginning of 2021[4]and finalized in 2022.[3]The software isopen sourceand published under theLGPLv2.1 license.[5]The specification is freely available.[2]
Because the colour adds a third dimension to the two-dimensional matrix, a JAB Code can contain more information in the same area than two-colour (black and white) codes; a four-colour code doubles the amount of data that can be stored, and an eight-colour code triples it. This increases the chances the barcode can store an entire message, rather than just partial data with a reference to a full message somewhere else (such as a link to a website), which would eliminate the need for additional always-available infrastructure beyond the printed barcode itself. It may be used todigitally signencrypted digital versions of printed legal documents, contracts, certificates (e.g., diplomas, training), and medical prescriptions or to provide product authenticity assurance, increasing protection against counterfeits.[3]
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https://en.wikipedia.org/wiki/JAB_Code
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PDF417is a stacked linearbarcodeformat used in a variety of applications such as transport, identification cards, and inventory management. "PDF" stands forPortable Data File, while "417" signifies that each pattern in the code consists of 4 bars and spaces in a pattern that is 17 units (modules) long.
The PDF417 symbology was invented by Dr. Ynjiun P. Wang atSymbol Technologiesin 1991.[1]It is defined in ISO 15438.
The PDF417 bar code (also called asymbol) consists of 3 to 90 rows, each of which is like a small linear bar code. Each row has:
All rows are the same width; each row has the same number of codewords.
PDF417 uses abase929 encoding. Each codeword represents a number from 0 to 928.
The codewords are represented by patterns of dark (bar) and light (space) regions. Each of these patterns contains four bars and four spaces (where the 4 in the name comes from). The total width is 17 times the width of the narrowest allowed vertical bar (the X dimension); this is where the 17 in the name comes from. Each pattern starts with a bar and ends with a space.
The row height must be at least 3 times the minimum width: Y ≥ 3 X.[2]: 5.8.2
There are three distinct bar–space patterns used to represent each codeword. These patterns are organized into three groups known asclusters. The clusters are labeled 0, 3, and 6. No bar–space pattern is used in more than one cluster. The rows of the symbol cycle through the three clusters, so row 1 uses patterns from cluster 0, row 2 uses cluster 3, row 3 uses cluster 6, and row 4 again uses cluster 0.
Which cluster can be determined by an equation:[2]: 5.3.1
WhereKis the cluster number and thebirefer to the width of thei-th black bar in the symbol character (inXunits).
Alternatively:[2]: 76–78
WhereEiis thei-th edge-to-next-same-edge distance. Odd indices are the leading edge of a bar to the leading edge of the next bar; even indices are for the trailing edges.
One purpose of the three clusters is to determine which row (mod 3) the codeword is in. The clusters allow portions of the symbol to be read using a single scan line that may be skewed from the horizontal.[2]: 5.11.1For instance, the scan might start on row 6 at the start of the row but end on row 10. At the beginning of the scan, the scanner sees the constant start pattern, and then it sees symbols in cluster 6. When the skewed scan straddles rows 6 and 7, then the scanner sees noise. When the scan is on row 7, the scanner sees symbols in cluster 0. Consequently, the scanner knows the direction of the skew. By the time the scanner reaches the right, it is on row 10, so it sees cluster 0 patterns. The scanner will also see a constant stop pattern.
Of the 929 available code words, 900 are used for data, and 29 for special functions, such as shifting between major modes. The three major modes encode different types of data in different ways, and can be mixed as necessary within a single bar code:
When the PDF417 symbol is created, from 2 to 512 error detection and correction codewords are added. PDF417 usesReed–Solomon error correction. When the symbol is scanned, the maximum number of corrections that can be made is equal to the number of codewords added, but the standard recommends that two codewords be held back to ensure reliability of the corrected information.
PDF417 is a stacked barcode that can be read with a simple linear scan being swept over the symbol.[3]Those linear scans need the left and right columns with the start and stop code words. Additionally, the scan needs to know what row it is scanning, so each row of the symbol must also encode its row number. Furthermore, the reader's line scan won't scan just a row; it will typically start scanning one row, but then cross over to a neighbor and possibly continuing on to cross successive rows. In order to minimize the effect of these crossings, the PDF417 modules are tall and narrow — the height is typically three times the width. Also, each code word must indicate which row it belongs to so crossovers, when they occur, can be detected. The code words are also designed to be delta-decodable, so some code words are redundant. Each PDF data code word represents about 10 bits of information (log2(900) ≈ 9.8), but the printed code word (character) is 17 modules wide. Including a height of 3 modules, a PDF417 code word takes 51 square modules to represent 10 bits. That area does not count other overhead such as the start, stop, row, format, and ECC information.
Other 2D codes, such asDataMatrixandQR, are decoded with image sensors instead of uncoordinated linear scans. Those codes still need recognition and alignment patterns, but they do not need to be as prominent. An 8 bit code word will take 8 square modules (ignoring recognition, alignment, format, and ECC information).
In practice, a PDF417 symbol takes about four times the area of a DataMatrix or QR Code.[4]
In addition to features typical of two dimensional bar codes, PDF417's capabilities include:
The introduction of the ISO/IEC document states:[2]
Manufacturers of bar code equipment and users of bar code technology require publicly available standard symbology specifications to which they can refer when developing equipment and application standards. It is the intent and understanding of ISO/IEC that the symbology presented in this International Standard is entirely in the public domain and free of all user restrictions, licences and fees.
PDF417 is used in many applications by both commercial and government organizations. PDF417 is one of the formats (along withData Matrix) that can be used to printpostageaccepted by theUnited States Postal Service. PDF417 is also used by the airline industry'sBar Coded Boarding Pass(BCBP) standard as the 2D bar code symbolism for paper boarding passes. PDF417 is the standard selected by theDepartment of Homeland Securityas the machine readable zone technology forRealIDcompliantdriver licensesand state issued identification cards. PDF417 barcodes are also included onvisasand border crossing cards issued by theState of Israel.
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https://en.wikipedia.org/wiki/PDF417
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QRpediais amobile Web-based system which usesQR codesto deliverWikipediaarticles to users, in their preferred language.[1][2][3]A typical use is onmuseum labels, linking to Wikipedia articles about the exhibited object. QR codes can easily be generated to link directly to anyUniform Resource Identifier(URI), but the QRpedia system adds further functionality. It is owned and operated by a subsidiary ofWikimedia UK(WMUK).
QRpedia was conceived by Roger Bamkin, a Wikipedia volunteer, coded byTerence Eden, and unveiled in April 2011. It is in use at museums and other institutions in countries includingAustralia,Bulgaria, theCzech Republic,Estonia,Malaysia,North Macedonia,Spain,India, theUnited Kingdom,Germany,South Africa,Sweden,Ukraine[4]and theUnited States. The project's source code is freely reusable under theMIT License.
When a user scans a QRpedia QR code on theirmobile device, the device decodes the QR code into aUniform Resource Locator(URL) using thedomain name"languagecode.qrwp.org" and whosepath(final part) is the title of a Wikipedia article, and sends a request for the article specified in the URL to the QRpediaweb server. It also transmits the language setting of the device.[5]
The QRpedia server then uses Wikipedia'sAPI[1]to determine whether there is a version of the specified Wikipedia article in the language used by the device, and if so, returns it in a mobile-friendly format.[5]If there is no version of the article available in the preferred language, then the QRpedia server offers a choice of the available languages, or aGoogle translation.
In this way, one QRcode can deliver the same article in many languages,[5]even when the museum is unable to make its own translations. QRpedia also records usage statistics.[5][6]
QRpedia was conceived by Roger Bamkin,[1][7]a Wikipedia volunteer, and Terence Eden,[1]a mobile web consultant,[8]and was unveiled on 9 April 2011[1][9]atDerby Museum and Art Gallery'sBackstage Passevent,[1][8]part of the "GLAM/Derby" collaboration between the museum and Wikipedia,[10]during which over 1,200 Wikipedia articles, in several languages, were also created.[11]The project's name is aportmanteauword, combining the initials "QR" from "QR (Quick Response) code" and "pedia" from "Wikipedia".[12]
The project's source code is freely reusable under the MIT License.[13]
Though created in the United Kingdom, QRpedia can be used in any location as long as the user's phone or tablet has a data signal (or remembers URLs until a signal is available) and is or has been in use at venues including:
QRpedia also has uses outside of such institutions. For example, theOccupy movementhave used it on campaign posters.[24]
In January 2012, QRpedia was one of four projects (from 79 entrants) declared the most innovative mobile companies in the UK of 2011 by the Smart UK Project, and thus chosen to compete atMobile World Congressin Barcelona, on 29 February 2012.[20]The criteria were "to be effective, easy to understand and with global potential and impact".[25]
A conflict of interest case involving QRpedia was identified as one of the "main incidents" leading to a 2012 review of the governance of Wikimedia UK (WMUK). The review found that the amount of time taken to resolve ownership caused the risk of outsiders perceiving a potential conflict of interest, and that Bamkin's acceptance of consultancy fees on projects (jointly funded by WMUK) involving QRpedia provided an opportunity for damage to the reputation of WMUK. This conflict of interest led to the resignation of WMUK trustee Joscelyn Upendran.[26]Shortly before her resignation on 31 August 2012, Upendran stated that "the charity has in effect agreed to take on responsibility [...] for a service that is 'co-owned' by a trustee", and suggested that "the conflict of interest may present a legal risk under charity and corporate law".[27]On 9 February 2013, WMUK announced that the intellectual property in QRpedia, and the qrpedia.org and qrwp.org domains, were to be transferred to the chapter at no cost.[28]On 12 February 2013, two QRpedia related domain names were registered on behalf of WMUK.[29]On 2 April 2013, WMUK announced that Roger Bamkin and Terence Eden were transferring ownership of QRpedia to Wikimedia UK.[30]On 16 November 2013, WMUK announced that the agreement for the transfer had been signed and the IP rights in QRpedia were held by Cultural Outreach Limited, a wholly owned subsidiary of WMUK, and that following the agreement, the transfer of the domain names was an administrative process that could begin immediately.[citation needed]
At least one Wikimedia chapter received letters alleging that QRpedia infringes various patents.[31]Though WMUK believes that this is not the case and that the risk of litigation is not high, Cultural Outreach Limited was set up to hold QRpedia, in order to shield WMUK should such a challenge arise.[32]
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https://en.wikipedia.org/wiki/QRpedia
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SnapTag, invented by SpyderLynk, is a2D mobile barcodealternative similar to aQR code, but that uses an icon or company logo and code ring rather than a square pattern of black dots.[1][2]
Similar to a QR code, SnapTags can be used to take consumers to a brand’s website, but can also facilitatemobile purchases,[3]coupon downloads,free samplerequests, video views, promotional entries,[4]Facebook Likes,PinterestPins,TwitterFollows, Posts and Tweets.[5]SnapTags offer back-end data mining capabilities.[6]
SnapTags can be used in Google's mobile Android operating system[7]and iOS devices (iPhone/iPod/iPad)[8]using The SnapTag Reader App or third party apps that have integrated the SnapTag Reader SDK. SnapTags can also be used by standard camera phones by taking a picture of the SnapTag and texting it to the designated short code or email address.[9][10]
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https://en.wikipedia.org/wiki/SnapTag
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ASPARQCodeis amatrix code(or two-dimensionalbar code)encodingstandard that is based on the physicalQR Codedefinition created by Japanese corporationDenso-Wave.
The QR Code standard as defined by Denso-Wave in ISO/IEC 18004 covers the physical encoding method of a binary data stream.[1]However, the Denso-Wave standard lacks an encoding standard for interpreting the data stream on the application layer for decoding URLs, phone numbers, and all other data types.NTT Docomohas established de facto standards for encoding some data types such as URLs, and contact information in Japan, but not all applications in other countries adhere to this convention as listed by the open-source project "zxing" for QR Code data types.[2][3]
The SPARQCode encoding standard specifies a convention for the following encoding data types.
The SPARQCode convention also recommends but does not require the inclusion of visual pictograms to denote the type of encoded data.
The use of the SPARQCode is free of any license. The termSPARQCodeitself is atrademarkof MSKYNET, but has chosen to open it to be royalty-free.[4]
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https://en.wikipedia.org/wiki/SPARQCode
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Touchatag(previouslyTikiTag[1]) was anRFIDservice for consumers, application developers and operators/enterprises created byAlcatel-Lucent. Consumers could use RFID tags to trigger what touchatag calledApplications, which could include opening a webpage, sending a text message, shutting down the computer, or running a custom application created through the software's API, via the application developer network. Touchatag applications were also compatible withNFCenabled phones like the Nokia 6212.[2][3]TikiTag was launched as anopen betaon October 1, 2008.[4][5]And it was rebranded to touchatag on February 15, 2009.[6]Touchatag also sold RFID hardware, like a starter package with 1 USB RFID reader and 10 RFID tags (stickers), for which the client software was compatible with Windows XP and Vista, along with Mac OS X 10.4 and up.[7]Touchatag was carried by Amazon.com, ThinkGeek, Firebox.com and getDigital.de[8]along with Touchatag's own Online Store.[9]Touchatag also marketed their products' underlying technology for enterprise and operator solutions. Touchatag announced an agreement with Belgacom PingPing on jointly developing the contactless market and announced a commercial pilot with Accor Services.[10]On June 27, 2012, the Touchatag team has announced the shutdown of the project.[11]inviting users to useIOTOPE"a similar open source Internet Of Things service" which itself has no apparent activity since November 2012.
Touchatag's core offering was the touchatag service, based on the "application correlation service" and allowed tag, reader and application management. For consumers, the web interface allowed to link RFID tags (and2D barcodetags, more preciselyQR Code) to applications. Application developers could use the correlation API to use the ACS functionalities to create contactless applications.[12]For businesses, this ACS was extended with an RFID/NFC tag and reader catalogue, and applications like loyalty, interactive advertising and couponing.[13]
The reader provided was an ACR122U Tag Reader, fromAdvanced Card Systems.[14]The tags shipped with the reader were MiFare Ultralight tags.[14]
Touchatag hardware was supported by its makers on Microsoft Windows and Mac OS X platforms, and required registration on the website to work. An unsupported application was also available for Linux platforms.[15]Like the Mac OS X application, the Linux application used PCSC-Lite for hardware access.
Many dead links since Touchatag ended its services
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https://en.wikipedia.org/wiki/Touchatag
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MaxiCodeis apublic domain, machine-readable symbol system originally created by theUnited Parcel Service(UPS) in 1992.[1]Suitable for tracking and managing the shipment of packages, it resembles anAztec CodeorQR code, but uses dots arranged in ahexagonal gridinstead ofsquare grid. MaxiCode has been standardised underISO/IEC16023.[2]
A MaxiCode symbol (internally called "Bird's Eye", "Target", "dense code", or "UPS code") appears as a 1-inch square, with abullseyein the middle, surrounded by a pattern of hexagonal dots. It can store about 93 characters of information, and up to 8 MaxiCode symbols can be chained together to convey more data. The centered symmetrical bullseye is useful in automatic symbol location regardless of orientation, and it allows MaxiCode symbols to be scanned even on a package traveling rapidly.
MaxiCode symbols using modes 2 and 3 include aStructured Carrier Messagecontaining key information about a package. This information is protected with a strongReed–Solomon error correctioncode, allowing it to be read even if a portion of the symbol is damaged. These fields include:
The structured portion of the message is stored in the inner area of the symbol, near the bull's-eye pattern. (In modes that do not include a structured portion, the inner area simply stores the beginning of the message.)
Irrespective of mode, a variable amount of application-specific information can be encoded in a MaxiCode symbol. This format of this additional data is not strictly defined, and amongst other information may include:
UPS labels use Mode 2 or Mode 3 MaxiCodes.
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https://en.wikipedia.org/wiki/MaxiCode
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Anauthenticatoris a means used to confirm a user's identity,[1][2]that is, to perform digital authentication. A person authenticates to a computer system or application by demonstrating that he or she has possession and control of an authenticator.[3][4]In the simplest case, the authenticator is a commonpassword.
Using the terminology of the NIST Digital Identity Guidelines,[3]the party to be authenticated is called theclaimantwhile the party verifying the identity of the claimant is called theverifier. When the claimant successfully demonstrates possession and control of one or more authenticators to the verifier through an established authentication protocol, the verifier is able to infer the claimant's identity.
Authenticators may be characterized in terms of secrets, factors, and physical forms.
Every authenticator is associated with at least one secret that the claimant uses to demonstrate possession and control of the authenticator. Since an attacker could use this secret to impersonate the user, an authenticator secret must be protected from theft or loss.
The type of secret is an important characteristic of the authenticator. There are three basic types of authenticator secret: a memorized secret and two types of cryptographic keys, either a symmetric key or a private key.
A memorized secret is intended to be memorized by the user. A well-known example of a memorized secret is the commonpassword, also called a passcode, apassphrase, or apersonal identification number(PIN).
An authenticator secret known to both the claimant and the verifier is called ashared secret. For example, a memorized secret may or may not be shared. A symmetric key is shared by definition. A private key is not shared.
An important type of secret that is both memorized and shared is the password. In the special case of a password, the authenticatoristhe secret.
A cryptographic authenticator is one that uses acryptographic key. Depending on the key material, a cryptographic authenticator may usesymmetric-key cryptographyorpublic-key cryptography. Both avoid memorized secrets, and in the case of public-key cryptography, there are noshared secretsas well, which is an important distinction.
Examples of cryptographic authenticators includeOATHauthenticators andFIDOauthenticators. The name OATH is an acronym from the words "Open AuTHentication" while FIDO stands for Fast IDentity Online. Both are the results of an industry-wide collaboration to develop an open reference architecture using open standards to promote the adoption of strong authentication.
By way of counterexample, a password authenticator isnota cryptographic authenticator. See the#Examplessection for details.
A symmetric key is a shared secret used to perform symmetric-key cryptography. The claimant stores their copy of the shared key in a dedicated hardware-based authenticator or a software-based authenticator implemented on a smartphone. The verifier holds a copy of the symmetric key.
A public-private key pair is used to perform public-key cryptography. The public key is known to (and trusted by) the verifier while the corresponding private key is bound securely to the authenticator. In the case of a dedicated hardware-based authenticator, the private key never leaves the confines of the authenticator.
An authenticator is something unique or distinctive to a user (something that one has), is activated by either aPIN(something that one knows), or is abiometric("something that is unique to oneself"). An authenticator that provides only one of these factors is called a single-factor authenticator whereas a multi-factor authenticator incorporates two or more factors. A multi-factor authenticator is one way to achievemulti-factor authentication. A combination of two or more single-factor authenticators is not a multi-factor authentication, yet may be suitable in certain conditions.
Authenticators may take a variety of physical forms (except for a memorized secret, which is intangible). One can, for example, hold an authenticator in one's hand or wear one on the face, wrist, or finger.[5][6][7]
It is convenient to describe an authenticator in terms of its hardware and software components. An authenticator is hardware-based or software-based depending on whether the secret is stored in hardware or software, respectively.
An important type of hardware-based authenticator is called a security key,[8]also called asecurity token(not to be confused withaccess tokens,session tokens, or other types of security tokens). A security key stores its secret in hardware, which prevents the secret from being exported. A security key is also resistant to malware since the secret is at no time accessible to software running on the host machine.
A software-based authenticator (sometimes called asoftware token) may be implemented on a general-purpose electronic device such as alaptop, atablet computer, or asmartphone. For example, a software-based authenticator implemented as amobile appon the claimant's smartphone is a type of phone-based authenticator. To prevent access to the secret, a software-based authenticator may use a processor'strusted execution environmentor aTrusted Platform Module(TPM) on the client device.
A platform authenticator is built into a particular client device platform, that is, it is implemented on device. In contrast, a roaming authenticator is a cross-platform authenticator that is implemented off device. A roaming authenticator connects to a device platform via a transport protocol such asUSB.
The following sections describe narrow classes of authenticators. For a more comprehensive classification, see the NIST Digital Identity Guidelines.[9]
To use an authenticator, the claimant must explicitly indicate their intent to authenticate. For example, each of the following gestures is sufficient to establish intent:
The latter is called a test of user presence (TUP). To activate a single-factor authenticator (something that one has), the claimant may be required to perform a TUP, which avoids unintended operation of the authenticator.
Apasswordis a secret that is intended to be memorized by the claimant and shared with the verifier. Password authentication is the process whereby the claimant demonstrates knowledge of the password by transmitting it over the network to the verifier. If the transmitted password agrees with the previously shared secret, user authentication is successful.
One-time passwords (OTPs) have been used since the 1980s.[citation needed]In 2004, an Open Authentication Reference Architecture for the secure generation of OTPs was announced at the annualRSA Conference.[10][11]TheInitiative for Open Authentication(OATH) launched a year later.[citation needed]Two IETF standards grew out of this work, theHMAC-based One-time Password (HOTP) algorithmand theTime-based One-time Password (TOTP) algorithmspecified by RFC 4226 and RFC 6238, respectively. By OATH OTP, we mean either HOTP or TOTP. OATH certifies conformance with the HOTP and TOTP standards.[12]
A traditional password (something that one knows) is often combined with a one-time password (something that one has) to provide two-factor authentication.[13]Both the password and the OTP are transmitted over the network to the verifier. If the password agrees with the previously shared secret, and the verifier can confirm the value of the OTP, user authentication is successful.
One-time passwords are generated on demand by a dedicated OATH OTP authenticator that encapsulates a secret that was previously shared with the verifier. Using the authenticator, the claimant generates an OTP using a cryptographic method. The verifier also generates an OTP using the same cryptographic method. If the two OTP values match, the verifier can conclude that the claimant possesses the shared secret.
A well-known example of an OATH authenticator isGoogle Authenticator,[14]a phone-based authenticator that implements both HOTP and TOTP.
A mobile push authenticator is essentially a native app running on the claimant's mobile phone. The app uses public-key cryptography to respond to push notifications. In other words, a mobile push authenticator is a single-factor cryptographic software authenticator. A mobile push authenticator (something that one has) is usually combined with a password (something that one knows) to provide two-factor authentication. Unlike one-time passwords, mobile push does not require a shared secret beyond the password.
After the claimant authenticates with a password, the verifier makes an out-of-band authentication request to a trusted third party that manages a public-key infrastructure on behalf of the verifier. The trusted third party sends a push notification to the claimant's mobile phone. The claimant demonstrates possession and control of the authenticator by pressing a button in the user interface, after which the authenticator responds with a digitally signed assertion. The trusted third party verifies the signature on the assertion and returns an authentication response to the verifier.
The proprietary mobile push authentication protocol runs on an out-of-band secondary channel, which provides flexible deployment options. Since the protocol requires an open network path to the claimant's mobile phone, if no such path is available (due to network issues, e.g.), the authentication process can not proceed.[13]
AFIDOUniversal 2nd Factor(U2F) authenticator (something that one has) is a single-factor cryptographic authenticator that is intended to be used in conjunction with an ordinary web password. Since the authenticator relies on public-key cryptography, U2F does not require an additional shared secret beyond the password.
To access a U2F authenticator, the claimant is required to perform a test of user presence (TUP), which helps prevent unauthorized access to the authenticator's functionality. In practice, a TUP consists of a simple button push.
A U2F authenticator interoperates with a conforming webuser agentthat implements the U2F JavaScript API.[15]A U2F authenticator necessarily implements the CTAP1/U2F protocol, one of the two protocols specified in the FIDOClient to Authenticator Protocol.[16]
Unlike mobile push authentication, the U2F authentication protocol runs entirely on the front channel. Two round trips are required. The first round trip is ordinary password authentication. After the claimant authenticates with a password, the verifier sends a challenge to a conforming browser, which communicates with the U2F authenticator via a custom JavaScript API. After the claimant performs the TUP, the authenticator signs the challenge and returns the signed assertion to the verifier via the browser.
To use a multi-factor authenticator, the claimant performs full user verification. The multi-factor authenticator (something that one has) is activated by aPIN(something that one knows), or abiometric(something that is unique to oneself"; e.g. fingerprint, face or voice recognition), or some other verification technique.[3],
To withdraw cash from anautomated teller machine(ATM), a bank customer inserts an ATM card into a cash machine and types a Personal Identification Number (PIN). The input PIN is compared to the PIN stored on the card's chip. If the two match, the ATM withdrawal can proceed.
Note that an ATM withdrawal involves a memorized secret (i.e., a PIN) but the true value of the secret is not known to the ATM in advance. The machine blindly passes the input PIN to the card, which compares the customer's input to the secret PIN stored on the card's chip. If the two match, the card reports success to the ATM and the transaction continues.
An ATM card is an example of a multi-factor authenticator. The card itself issomething that one haswhile the PIN stored on the card's chip is presumablysomething that one knows. Presenting the card to the ATM and demonstrating knowledge of the PIN is a kind of multi-factor authentication.
Secure Shell(SSH) is a client-server protocol that uses public-key cryptography to create a secure channel over the network. In contrast to a traditional password, an SSH key is a cryptographic authenticator. The primary authenticator secret is the SSH private key, which is used by the client to digitally sign a message. The corresponding public key is used by the server to verify the message signature, which confirms that the claimant has possession and control of the private key.
To avoid theft, the SSH private key (something that one has) may be encrypted using apassphrase(something that one knows). To initiate a two-factor authentication process, the claimant supplies the passphrase to the client system.
Like a password, the SSH passphrase is a memorized secret but that is where the similarity ends. Whereas a password is a shared secret that is transmitted over the network, the SSH passphrase is not shared, and moreover, use of the passphrase is strictly confined to the client system. Authentication via SSH is an example ofpasswordless authenticationsince it avoids the transmission of a shared secret over the network. In fact, SSH authentication does not require a shared secret at all.
The FIDO U2F protocol standard became the starting point for theFIDO2 Project, a joint effort between the World Wide Web Consortium (W3C) and the FIDO Alliance. Project deliverables include the W3C Web Authentication (WebAuthn) standard and the FIDOClient to Authenticator Protocol(CTAP).[17]Together WebAuthn and CTAP provide a strong authentication solution for the web.
A FIDO2 authenticator, also called a WebAuthn authenticator, uses public-key cryptography to interoperate with a WebAuthn client, that is, a conforming webuser agentthat implements the WebAuthnJavaScriptAPI.[18]The authenticator may be a platform authenticator, a roaming authenticator, or some combination of the two. For example, a FIDO2 authenticator that implements the CTAP2 protocol[16]is a roaming authenticator that communicates with a WebAuthn client via one or more of the following transport options:USB,near-field communication(NFC), orBluetooth Low Energy(BLE). Concrete examples of FIDO2 platform authenticators include Windows Hello[19]and theAndroid operating system.[20]
A FIDO2 authenticator may be used in either single-factor mode or multi-factor mode. In single-factor mode, the authenticator is activated by a simple test of user presence (e.g., a button push). In multi-factor mode, the authenticator (something that one has) is activated by either aPIN(something that one knows) or abiometric("something that is unique to oneself").
First and foremost, strong authentication begins withmulti-factor authentication. The best thing one can do to protect a personal online account is to enable multi-factor authentication.[13][21]There are two ways to achieve multi-factor authentication:
In practice, a common approach is to combine a password authenticator (something that one knows) with some other authenticator (something that one has) such as a cryptographic authenticator.
Generally speaking, acryptographic authenticatoris preferred over an authenticator that does not use cryptographic methods. All else being equal, a cryptographic authenticator that uses public-key cryptography is better than one that uses symmetric-key cryptography since the latter requires shared keys (which may be stolen or misused).
Again all else being equal, a hardware-based authenticator is better than a software-based authenticator since the authenticator secret is presumably better protected in hardware. This preference is reflected in the NIST requirements outlined in the next section.
NIST defines three levels of assurance with respect to authenticators. The highest authenticator assurance level (AAL3) requires multi-factor authentication using either a multi-factor authenticator or an appropriate combination of single-factor authenticators. At AAL3, at least one of the authenticators must be a cryptographic hardware-based authenticator. Given these basic requirements, possible authenticator combinations used at AAL3 include:
See the NIST Digital Identity Guidelines for further discussion of authenticator assurance levels.[9]
Like authenticator assurance levels, the notion of a restricted authenticator is a NIST concept.[3]The term refers to an authenticator with a demonstrated inability to resist attacks, which puts the reliability of the authenticator in doubt. Federal agencies mitigate the use a restricted authenticator by offering subscribers an alternative authenticator that is not restricted and by developing a migration plan in the event that a restricted authenticator is prohibited from use at some point in the future.
Currently, the use of thepublic switched telephone networkis restricted by NIST. In particular, the out-of-band transmission of one-time passwords (OTPs) via recorded voice messages orSMSmessages is restricted. Moreover, if an agency chooses to use voice- or SMS-based OTPs, that agency must verify that the OTP is being transmitted to a phone and not an IP address sinceVoice over IP(VoIP) accounts are not routinely protected with multi-factor authentication.[9]
It is convenient to use passwords as a basis for comparison since it is widely understood how to use a password.[22]On computer systems, passwords have been used since at least the early 1960s.[23][24]More generally, passwords have been used since ancient times.[25]
In 2012, Bonneau et al. evaluated two decades of proposals to replace passwords by systematically comparing web passwords to 35 competing authentication schemes in terms of their usability, deployability, and security.[26](The cited technical report is an extended version of the peer-reviewed paper by the same name.[27]) They found that most schemes do better than passwords on security whileeveryscheme does worse than passwords on deployability. In terms of usability, some schemes do better and some schemes do worse than passwords.
Google used the evaluation framework of Bonneau et al. to compare security keys to passwords and one-time passwords.[28]They concluded that security keys are more usable and deployable than one-time passwords, and more secure than both passwords and one-time passwords.
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Ahardware security module(HSM) is a physical computing device that safeguards and manages secrets (most importantlydigital keys), and performsencryptionand decryption functions fordigital signatures, strong authentication and other cryptographic functions.[1]These modules traditionally come in the form of a plug-in card or an external device that attaches directly to acomputerornetwork server. A hardware security module contains one or moresecure cryptoprocessorchips.[2][3]
HSMs may have features that provide tamper evidence such as visible signs of tampering or logging and alerting, or tamper resistance which makes tampering difficult without making the HSM inoperable, or tamper responsiveness such as deleting keys upon tamper detection.[4]Each module contains one or moresecure cryptoprocessorchips to prevent tampering andbus probing, or a combination of chips in a module that is protected by the tamper evident, tamper resistant, or tamper responsive packaging. A vast majority of existing HSMs are designed mainly to manage secret keys. Many HSM systems have means to securely back up the keys they handle outside of the HSM. Keys may be backed up in wrapped form and stored on acomputer diskor other media, or externally using a secure portable device like asmartcardor some othersecurity token.[5]
HSMs are used for real time authorization and authentication in critical infrastructure thus are typically engineered to support standard high availability models includingclustering, automatedfailover, and redundantfield-replaceable components.
A few of the HSMs available in the market have the capability to execute specially developed modules within the HSM's secure enclosure. Such an ability is useful, for example, in cases where special algorithms or business logic has to be executed in a secured and controlled environment. The modules can be developed in nativeC language, .NET,Java, or other programming languages.
Due to the critical role they play in securing applications and infrastructure, general purpose HSMs and/or the cryptographic modules are typically certified according to internationally recognized standards such asCommon Criteria(e.g. using Protection Profile EN 419 221-5, "Cryptographic Module for Trust Services") orFIPS 140(currently the 3rd version, often referred to as FIPS 140-3). Although the highest level ofFIPS 140security certification attainable is Security Level 4, most of the HSMs have Level 3 certification. In the Common Criteria system the highest EAL (Evaluation Assurance Level) is EAL7, most of the HSMs have EAL4+ certification. When used in financial payments applications, the security of an HSM is often validated against the HSM requirements defined by thePayment Card Industry Security Standards Council.[6]
A hardware security module can be employed in any application that uses digital keys. Typically, the keys would be of high value - meaning there would be a significant, negative impact to the owner of the key if it were compromised.
The functions of an HSM are:
HSMs are also deployed to managetransparent data encryptionkeys for databases and keys for storage devices such asdiskortape.[citation needed]
Some HSM systems are also hardwarecryptographic accelerators. They usually cannot beat the performance of hardware-only solutions for symmetric key operations. However, with performance ranges from 1 to 10,000 1024-bitRSAsignatures per second, HSMs can provide significant CPU offload for asymmetric key operations. Since theNational Institute of Standards and Technology(NIST) is recommending the use of 2,048 bit RSA keys from year 2010,[7]performance at longer key sizes has become more important. To address this issue, most HSMs now supportelliptic curve cryptography(ECC), which delivers stronger encryption with shorter key lengths.
InPKIenvironments, the HSMs may be used bycertification authorities(CAs) andregistration authorities(RAs) to generate, store, and handle asymmetric key pairs. In these cases, there are some fundamental features a device must have, namely:
On the other hand, device performance in a PKI environment is generally less important, in both online and offline operations, as Registration Authority procedures represent the performance bottleneck of the Infrastructure.
Specialized HSMs are used in the payment card industry. HSMs support both general-purpose functions and specialized functions required to process transactions and comply with industry standards. They normally do not feature a standardAPI.
Typical applications are transaction authorization and payment card personalization, requiring functions such as:
The major organizations that produce and maintain standards for HSMs on the banking market are thePayment Card Industry Security Standards Council,ANS X9, andISO.
Performance-critical applications that have to useHTTPS(SSL/TLS), can benefit from the use of an SSL Acceleration HSM by moving the RSA operations, which typically requires several large integer multiplications, from the host CPU to the HSM device. Typical HSM devices can perform about 1 to 10,000 1024-bit RSA operations/second.[8][9]Some performance at longer key sizes is becoming increasingly important.
An increasing number of registries use HSMs to store the key material that is used to sign largezonefiles.OpenDNSSECis an open-source tool that manages signing DNSzone files.
On January 27, 2007,ICANNandVerisign, with support from theU.S. Department of Commerce, started deployingDNSSECforDNS root zones.[10]Root signature details can be found on the Root DNSSEC's website.[11]
Blockchaintechnology depends on cryptographic operations. Safeguarding private keys is essential to maintain the security of blockchain processes that utilize asymmetric cryptography. The private keys are often stored in acryptocurrency walletlike the hardware wallet in the image.
The synergy between HSMs and blockchain is mentioned in several papers, emphasizing their role in securing private keys and verifying identity, e.g. in contexts such as blockchain-driven mobility solutions.[12][13]
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Initiative for Open Authentication(OATH) is an industry-wide collaboration to develop an openreference architectureusingopen standardsto promote the adoption of strong authentication. It has close to thirty coordinating and contributing members and is proposing standards for a variety of authentication technologies, with the aim of lowering costs and simplifying their functions.
The nameOATHis an acronym from the phrase "open authentication", and is pronounced as the English word "oath".[1]
OATH is not related toOAuth, an open standard forauthorization, however, most logging systems employ a mixture of both.
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Asoftware token(a.k.a.soft token) is a piece of atwo-factor authenticationsecurity device that may be used to authorize the use of computer services.[1]Software tokens are stored on a general-purpose electronic device such as adesktop computer,laptop,PDA, ormobile phoneand can be duplicated. (Contrasthardware tokens, where the credentials are stored on a dedicated hardware device and therefore cannot be duplicated — absent physical invasion of the device)
Because software tokens are something one does not physically possess, they are exposed to unique threats based on duplication of the underlying cryptographic material - for example,computer virusesandsoftwareattacks. Both hardware and software tokens are vulnerable to bot-basedman-in-the-middle attacks, or to simplephishingattacks in which theone-time passwordprovided by the token is solicited, and then supplied to the genuine website in a timely manner. Software tokens do have benefits: there is no physical token to carry, they do not containbatteriesthat will run out, and they are cheaper than hardware tokens.[2]
There are two primary architectures for software tokens:shared secretandpublic-key cryptography.
For a shared secret, anadministratorwill typically generate aconfiguration filefor each end-user. The file will contain a username, apersonal identification number, and thesecret. This configuration file is given to the user.
The shared secret architecture is potentially vulnerable in a number of areas. The configuration file can be compromised if it is stolen and the token is copied. With time-based software tokens, it is possible to borrow an individual'sPDAor laptop, set the clock forward, and generate codes that will be valid in the future. Any software token that uses shared secrets and stores the PIN alongside the shared secret in a software client can be stolen and subjected to offline attacks. Shared secret tokens can be difficult to distribute, since each token is essentially a different piece of software. Each user must receive a copy of the secret, which can create time constraints.
Some newer software tokens rely onpublic-key cryptography, or asymmetric cryptography. Thisarchitectureeliminates some of the traditional weaknesses of software tokens, but does not affect their primary weakness (ability to duplicate). A PIN can be stored on a remote authenticationserverinstead of with the token client, making a stolen software token no good unless the PIN is known as well. However, in the case of a virus infection, the cryptographic material can be duplicated and then the PIN can be captured (via keylogging or similar) the next time the user authenticates. If there are attempts made to guess the PIN, it can be detected and logged on the authentication server, which can disable the token. Using asymmetric cryptography also simplifies implementation, since the token client can generate its own key pair and exchange public keys with the server.
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Botanis aBSD-licensedcryptographicand TLS library written inC++11. It provides a wide variety of cryptographic algorithms, formats, and protocols, e.g.SSL and TLS. It is used in theMonotonedistributed revision controlprogram, theOpenDNSSECsystem, andISC's Kea DHCP serveramong other projects.
The project was originally calledOpenCL, a name now used byApple Inc.andKhronos Groupfor aheterogeneous system programming framework. It was renamed Botan in 2002.[2]
In 2007, the GermanFederal Office for Information SecuritycontractedFlexSecure GmbHto add an implementation ofCard Verifiable CertificatesforePassportsto Botan; the modified version of Botan was released under the name InSiTo.[3]
Starting in 2015, the German Federal Office for Information Security funded a project, which included improving the documentation, test suite and feature set of Botan, culminating in 2017, when it was evaluated and recommended as a library suitable for "applications with increased security requirements".[4]
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multiOTPis an open source PHP class, a command line tool, and a web interface that can be used to provide an operating-system-independent, strongauthenticationsystem. multiOTP isOATH-certified since version 4.1.0 and is developed under theLGPLlicense. Starting with version 4.3.2.5, multiOTP open source is also available as a virtual appliance—as a standard OVA file, a customized OVA file with open-vm-tools, and also as avirtual machinedownloadable file that can run on Microsoft'sHyper-V, a common nativehypervisorin Windows computers.[jargon]
AQR codeis generated automatically when printing the user-configuration page.
Spyware, viruses and other hacking technologies or bugs (such asHeartbleed) are regularly used to steal passwords. If a strongtwo-factorauthentication system is used, the stolen passwords cannot be stored and later used because eachone-time passwordis valid for only one authentication session, and will fail if tried a second time.[1]
multiOTP is a PHP class library. The class can be used with any PHP application using a PHP version of 5.3.0 or higher. The multiOTP library is provided as an all-in-one self-contained file that requires no other includes. If the strong authentication needs to be done from a hardware device instead of an Internet application, a request will go through a RADIUS server which will call the multiOTP command line tool. The implementation is light enough in order to work on limited computers, such as theRaspberry Pi.
For Windows, the multiOTP library is provided with a pre-configured RADIUS server (freeradius) which can be installed as a service. A pre-configured web service (based on mongoose) can also be installed as a service and is needed if we want to use the multiOTP library in a client/server configuration.
Under Linux, the readme.txt file provided with the library indicates what should be done in order to configure the RADIUS server and the web service.
All necessary files and instructions are also provided to make a strong authentication device using a Raspberry Pi nano-computer.
Since version 4.3.2.5, ready to use virtual appliance is provided in standard OVA format, with open-vm-tools integrated and also in Hyper-V format.
The client can strongly authenticate on an application or a device using different methods:
multiOTP isInitiative For Open Authenticationcertified for HOTP and TOTP and currently supports the following algorithms and RFCs:
The multiOTP class provides strong authentication functionality and can be used in different strong authentication situations:
Several free projects use the library:
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The following is a generalcomparison of OTP applicationsthat are used to generateone-time passwordsfortwo-factor authentication(2FA) systems using thetime-based one-time password(TOTP) or theHMAC-based one-time password(HOTP) algorithms.
by 2Stable[45]
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Authentication(fromGreek:αὐθεντικόςauthentikos, "real, genuine", from αὐθέντηςauthentes, "author") is the act of proving anassertion, such as theidentityof a computer system user. In contrast withidentification, the act of indicating a person or thing's identity, authentication is the process of verifying that identity.[1][2]
Authentication is relevant to multiple fields. Inart,antiques, andanthropology, a common problem is verifying that a given artifact was produced by a certain person, or in a certain place (i.e. to assert that it is notcounterfeit), or in a given period of history (e.g. by determining the age viacarbon dating). Incomputer science, verifying a user's identity is often required to allow access to confidential data or systems.[3]It might involve validating personalidentity documents.
Authentication can be considered to be of three types:
Thefirsttype of authentication is accepting proof of identity given by a credible person who has first-hand evidence that the identity is genuine. When authentication is required of art or physical objects, this proof could be a friend, family member, or colleague attesting to the item's provenance, perhaps by having witnessed the item in its creator's possession. With autographed sportsmemorabilia, this could involve someone attesting that they witnessed the object being signed. A vendor selling branded items implies authenticity, while they may not have evidence that every step in the supply chain was authenticated.
Thesecondtype of authentication is comparing the attributes of the object itself to what is known about objects of that origin. For example, an art expert might look for similarities in the style of painting, check the location and form of a signature, or compare the object to an old photograph. Anarchaeologist, on the other hand, might use carbon dating to verify the age of an artifact, do a chemical andspectroscopicanalysis of the materials used, or compare the style of construction or decoration to other artifacts of similar origin. The physics of sound and light, and comparison with a known physical environment, can be used to examine the authenticity of audio recordings, photographs, or videos. Documents can be verified as being created on ink or paper readily available at the time of the item's implied creation.
Attribute comparison may be vulnerable toforgery. In general, it relies on the facts that creating a forgery indistinguishable from a genuine artifact requires expert knowledge, that mistakes are easily made, and that the amount of effort required to do so is considerably greater than the amount of profit that can be gained from the forgery.
In art and antiques, certificates are of great importance for authenticating an object of interest and value. Certificates can, however, also be forged, and the authentication of these poses a problem. For instance, the son ofHan van Meegeren, the well-known art-forger, forged the work of his father and provided a certificate for its provenance as well.
Criminal and civil penalties forfraud,forgery, and counterfeiting can reduce the incentive for falsification, depending on the risk of getting caught.
Currency and other financial instruments commonly use this second type of authentication method. Bills, coins, andchequesincorporate hard-to-duplicate physical features, such as fine printing or engraving, distinctive feel, watermarks, and holographic imagery, which are easy for trained receivers to verify.
Thethirdtype of authentication relies on documentation or other external affirmations. In criminal courts, therules of evidenceoften require establishing thechain of custodyof evidence presented. This can be accomplished through a written evidence log, or by testimony from the police detectives and forensics staff that handled it. Some antiques are accompanied by certificates attesting to their authenticity. Signed sports memorabilia is usually accompanied by a certificate of authenticity. These external records have their own problems of forgery andperjuryand are also vulnerable to being separated from the artifact and lost.
Consumer goodssuch as pharmaceuticals,[4]perfume, and clothing can use all forms of authentication to prevent counterfeit goods from taking advantage of a popular brand's reputation. As mentioned above, having an item for sale in a reputable store implicitly attests to it being genuine, the first type of authentication. The second type of authentication might involve comparing the quality and craftsmanship of an item, such as an expensive handbag, to genuine articles. The third type of authentication could be the presence of atrademarkon the item, which is a legally protected marking, or any other identifying feature which aids consumers in the identification of genuine brand-name goods. With software, companies have taken great steps to protect from counterfeiters, including adding holograms, security rings, security threads and color shifting ink.[5]
Counterfeit products are often offered to consumers as being authentic.Counterfeit consumer goods, such as electronics, music, apparel, andcounterfeit medications, have been sold as being legitimate. Efforts to control thesupply chainand educate consumers help ensure that authentic products are sold and used. Evensecurity printingon packages, labels, and nameplates, however, is subject to counterfeiting.[6]
In their anti-counterfeiting technology guide,[7]theEUIPOObservatory on Infringements of Intellectual Property Rights categorizes the main anti-counterfeiting technologies on the market currently into five main categories: electronic, marking, chemical and physical, mechanical, and technologies for digital media.[8]
Products or their packaging can include a variableQR Code. A QR Code alone is easy to verify but offers a weak level of authentication as it offers no protection against counterfeits unless scan data is analyzed at the system level to detect anomalies.[9]To increase the security level, the QR Code can be combined with adigital watermarkorcopy detection patternthat are robust to copy attempts and can be authenticated with a smartphone.
Asecure key storage devicecan be used for authentication in consumer electronics, network authentication, license management, supply chain management, etc. Generally, the device to be authenticated needs some sort of wireless or wired digital connection to either a host system or a network. Nonetheless, the component being authenticated need not be electronic in nature as an authentication chip can be mechanically attached and read through a connector to the host e.g. an authenticated ink tank for use with a printer. For products and services that these secure coprocessors can be applied to, they can offer a solution that can be much more difficult to counterfeit than most other options while at the same time being more easily verified.[2]
Packaging and labeling can be engineered to help reduce the risks of counterfeit consumer goods or the theft and resale of products.[10][11]Some package constructions are more difficult to copy and some have pilfer indicating seals. Counterfeit goods, unauthorized sales (diversion), material substitution and tampering can all be reduced with these anti-counterfeiting technologies. Packages may include authentication seals and usesecurity printingto help indicate that the package and contents are not counterfeit; these too are subject to counterfeiting. Packages also can include anti-theft devices, such as dye-packs,RFIDtags, orelectronic article surveillance[12]tags that can be activated or detected by devices at exit points and require specialized tools to deactivate. Anti-counterfeiting technologies that can be used with packaging include:
In literacy, authentication is a readers’ process of questioning the veracity of an aspect of literature and then verifying those questions via research. The fundamental question for authentication of literature is – Does one believe it? Related to that, an authentication project is therefore a reading and writing activity in which students document the relevant research process.[13]It builds students' critical literacy. The documentation materials for literature go beyond narrative texts and likely include informational texts, primary sources, and multimedia. The process typically involves both internet and hands-on library research. When authenticating historical fiction in particular, readers consider the extent that the major historical events, as well as the culture portrayed (e.g., the language, clothing, food, gender roles), are believable for the period.[3]Literary forgerycan involve imitating the style of a famous author. If an originalmanuscript, typewritten text, or recording is available, then the medium itself (or its packaging – anything from a box toe-mail headers) can help prove or disprove the authenticity of the document. However, text, audio, and video can be copied into new media, possibly leaving only the informational content itself to use in authentication. Various systems have been invented to allow authors to provide a means for readers to reliably authenticate that a given message originated from or was relayed by them. These involve authentication factors like:
The opposite problem is the detection ofplagiarism, where information from a different author is passed off as a person's own work. A common technique for proving plagiarism is the discovery of another copy of the same or very similar text, which has different attribution. In some cases, excessively high quality or a style mismatch may raise suspicion of plagiarism.
The process of authentication is distinct from that ofauthorization. Whereas authentication is the process of verifying that "you are who you say you are", authorization is the process of verifying that "you are permitted to do what you are trying to do". While authorization often happens immediately after authentication (e.g., when logging into a computer system), this does not mean authorization presupposes authentication: an anonymous agent could be authorized to a limited action set.[14]Similarly, the establishment of the authorization can occur long before theauthorizationdecision occurs.
A user can be given access to secure systems based on user credentials that imply authenticity.[15]A network administrator can give a user apassword, or provide the user with a key card or other access devices to allow system access. In this case, authenticity is implied but not guaranteed.
Most secure internet communication relies on centralized authority-based trust relationships, such as those used inHTTPS, where publiccertificate authorities(CAs) vouch for the authenticity of websites. This same centralized trust model underpins protocols like OIDC (OpenID Connect) where identity providers (e.g.,Google) authenticate users on behalf of relying applications. In contrast, decentralized peer-based trust, also known as aweb of trust, is commonly used for personal services such as secure email or file sharing. In systems likePGP, trust is established when individuals personally verify and sign each other’s cryptographic keys, without relying on a central authority.
These systems usecryptographicprotocolsthat, in theory, are not vulnerable tospoofingas long as the originator’s private key remains uncompromised. Importantly, even if the key owner is unaware of a compromise, the cryptographic failure still invalidates trust. However, while these methods are currently considered secure, they are not provably unbreakable—future mathematical or computational advances (such asquantum computingor new algorithmic attacks) could expose vulnerabilities. If that happens, it could retroactively undermine trust in past communications or agreements. For example, adigitally signedcontractmight be challenged if the signature algorithm is later found to be insecure..[citation needed]
The ways in which someone may be authenticated fall into three categories, based on what is known as the factors of authentication: something the user knows, something the user has, and something the user is. Each authentication factor covers a range of elements used to authenticate or verify a person's identity before being granted access, approving a transaction request, signing a document or other work product, granting authority to others, and establishing a chain of authority.
Security research has determined that for a positive authentication, elements from at least two, and preferably all three, factors should be verified.[16][17]The three factors (classes) and some of the elements of each factor are:
As the weakest level of authentication, only a single component from one of the three categories of factors is used to authenticate an individual's identity. The use of only one factor does not offer much protection from misuse or malicious intrusion. This type of authentication is not recommended for financial or personally relevant transactions that warrant a higher level of security.[21]
Multi-factor authentication involves two or more authentication factors (something you know, something you have, or something you are). Two-factor authentication is a special case of multi-factor authentication involving exactly two factors.[21]
For example, using a bank card (something the user has) along with a PIN (something the user knows) provides two-factor authentication. Business networks may require users to provide a password (knowledge factor) and a pseudorandom number from a security token (ownership factor). Access to a very-high-security system might require amantrapscreening of height, weight, facial, and fingerprint checks (several inherence factor elements) plus a PIN and a day code (knowledge factor elements),[22]but this is still a two-factor authentication.
The United States government'sNational Information Assurance Glossarydefines strong authentication as a layered authentication approach relying on two or more authenticators to establish the identity of an originator or receiver of information.[23]
The European Central Bank (ECB) has defined strong authentication as "a procedure based on two or more of the three authentication factors". The factors that are used must be mutually independent and at least one factor must be "non-reusable and non-replicable", except in the case of an inherence factor and must also be incapable of being stolen off the Internet. In the European, as well as in the US-American understanding, strong authentication is very similar to multi-factor authentication or 2FA, but exceeding those with more rigorous requirements.[21][24]
TheFIDO Alliancehas been striving to establish technical specifications for strong authentication.[25]
Conventional computer systems authenticate users only at the initial log-in session, which can be the cause of a critical security flaw. To resolve this problem, systems need continuous user authentication methods that continuously monitor and authenticate users based on some biometric trait(s). A study used behavioural biometrics based on writing styles as a continuous authentication method.[26][27]
Recent research has shown the possibility of using smartphones sensors and accessories to extract some behavioral attributes such as touch dynamics,keystroke dynamicsandgait recognition.[28]These attributes are known as behavioral biometrics and could be used to verify or identify users implicitly and continuously on smartphones. The authentication systems that have been built based on these behavioral biometric traits are known as active or continuous authentication systems.[29][27]
The term digital authentication, also known aselectronic authenticationor e-authentication, refers to a group of processes where the confidence for user identities is established and presented via electronic methods to an information system. The digital authentication process creates technical challenges because of the need to authenticate individuals or entities remotely over a network.
The AmericanNational Institute of Standards and Technology(NIST) has created a generic model for digital authentication that describes the processes that are used to accomplish secure authentication:
The authentication of information can pose special problems with electronic communication, such as vulnerability toman-in-the-middle attacks, whereby a third party taps into the communication stream, and poses as each of the two other communicating parties, in order to intercept information from each. Extra identity factors can be required to authenticate each party's identity.
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Electronic authenticationis the process of establishing confidence in user identities electronically presented to aninformation system.[1]Digital authentication,ore-authentication,may be used synonymously when referring to theauthenticationprocess that confirms or certifies a person's identity and works. When used in conjunction with anelectronic signature, it can provide evidence of whetherdatareceived has been tampered with after being signed by its original sender. Electronic authentication can reduce the risk offraudandidentity theftby verifying that a person is who they say they are when performing transactions online.[2]
Various e-authentication methods can be used to authenticate a user's identify ranging from apasswordto higher levels of security that utilizemulti-factor authentication(MFA).[3]Depending on the level of security used, the user might need to prove his or her identity through the use ofsecurity tokens, challenge questions, or being in possession of a certificate from a third-party certificate authority that attests to their identity.[4]
The AmericanNational Institute of Standards and Technology(NIST) has developed a generic electronic authentication model[5]that provides a basic framework on how the authentication process is accomplished regardless of jurisdiction or geographic region. According to this model, the enrollment process begins with an individual applying to aCredential Service Provider(CSP). The CSP will need to prove the applicant's identity before proceeding with the transaction.[6]Once the applicant's identity has been confirmed by the CSP, he or she receives the status of "subscriber", is given anauthenticator, such as a token and a credential, which may be in the form of a username.
The CSP is responsible for managing the credential along with the subscriber's enrollment data for the life of the credential. The subscriber will be tasked with maintaining the authenticators. An example of this is when a user normally uses a specific computer to do theironline banking. If he or she attempts to access their bank account from another computer, the authenticator will not be present. In order to gain access, the subscriber would need to verify their identity to the CSP, which might be in the form of answering a challenge question successfully before being given access.[4]
Well-maintained health records can help doctors and hospitals know the targeted patient's important medical conditions before conducting any therapy.
Therefore, to safely establish and manage personal health records for each individual during his/her lifetime within the electronic form has gradually become an interesting topic for individual citizens and social welfare departments.
As this data is private by nature, electronic authorization helps to ensure that only permitted parties can access the medical data.
The need for authentication has been prevalent throughout history. In ancient times, people would identify each other through eye contact and physical appearance. TheSumeriansin ancientMesopotamiaattested to the authenticity of their writings by using seals embellished with identifying symbols. As time moved on, the most common way to provide authentication would be the handwritten signature.[2]
There are three generally accepted factors that are used to establish a digital identity for electronic authentication, including:
Out of the three factors, the biometric factor is the most convenient and convincing to prove an individual's identity, but it is the most expensive to implement. Each factor has its weaknesses; hence, reliable and strong authentication depends on combining two or more factors. This is known asmulti-factor authentication,[2]of which two-factor authentication and two-step verification are subtypes.
Multi-factor authentication can still be vulnerable to attacks, includingman-in-the-middle attacksand Trojan attacks.[7]
Tokens generically are something the claimant possesses and controls that may be used to authenticate the claimant's identity. In e-authentication, the claimant authenticates to a system or application over a network. Therefore, a token used for e-authentication is a secret and the token must be protected. The token may, for example, be a cryptographic key, that is protected by encrypting it under a password. An impostor must steal the encrypted key and learn the password to use the token.
Passwords and PINs are categorized as "something you know" method. A combination of numbers, symbols, and mixed cases are considered to be stronger than all-letter password. Also, the adoption of Transport Layer Security (TLS) or Secure Socket Layer (SSL) features during the information transmission process will as well create an encrypted channel for data exchange and to further protect information delivered. Currently, most security attacks target on password-based authentication systems.[8]
This type of authentication has two parts. One is a public key, the other is a private key. A public key is issued by a Certification Authority and is available to any user or server. A private key is known by the user only.[9]
The user shares a unique key with an authentication server. When the user sends a randomly generated message (the challenge) encrypted by the secret key to the authentication server, if the message can be matched by the server using its shared secret key, the user is authenticated.
When implemented together with the password authentication, this method also provides a possible solution fortwo-factor authenticationsystems.[10]
The user receives password by reading the message in the cell phone, and types back the password to complete the authentication.Short Message Service(SMS) is very effective when cell phones are commonly adopted. SMS is also suitable against man-in-the-middle (MITM) attacks, since the use of SMS does not involve the Internet.[11]
Biometric authentication is the use of unique physical attributes and body measurements as the intermediate for better identification and access control. Physical characteristics that are often used for authentication include fingerprints,voice recognition,face recognition, and iris scans because all of these are unique to every individual. Traditionally, biometric authentication based on token-based identification systems, such as passport, and nowadays becomes one of the most secure identification systems to user protections. A new technological innovation which provides a wide variety of either behavioral or physical characteristics which are defining the proper concept of biometric authentication.[12]
Digital identity authentication refers to the combined use of device, behavior, location and other data, including email address, account and credit card information, to authenticate online users in real time. For example, recent work have explored how to exploitbrowser fingerprintingas part of a multi-factor authentication scheme.[13]
Paper credentials are documents that attest to the identity or other attributes of an individual or entity called the subject of the credentials. Some common paper credentials include passports,birth certificates, driver's licenses, and employee identity cards. The credentials themselves are authenticated in a variety of ways: traditionally perhaps by a signature or a seal, special papers and inks, high quality engraving, and today by more complex mechanisms, such as holograms, that make the credentials recognizable and difficult to copy or forge. In some cases, simple possession of the credentials is sufficient to establish that the physical holder of the credentials is indeed the subject of the credentials.
More commonly, the credentials contain biometric information such as the subject's description, a picture of the subject or the handwritten signature of the subject that can be used to authenticate that the holder of the credentials is indeed the subject of the credentials. When these paper credentials are presented in-person, authenticationbiometricscontained in those credentials can be checked to confirm that the physical holder of the credential is the subject.
Electronic identity credentials bind a name and perhaps other attributes to a token. There are a variety ofelectronic credentialtypes in use today, and new types of credentials are constantly being created (eID,electronic voter ID card, biometric passports, bank cards, etc.) At a minimum, credentials include identifying information that permits recovery of the records of the registration associated with the credentials and a name that is associated with the subscriber.[citation needed]
In any authenticated on-line transaction, the verifier is the party that verifies that the claimant has possession and control of the token that verifies his or her identity. A claimant authenticates his or her identity to a verifier by the use of a token and an authentication protocol. This is called Proof of Possession (PoP). Many PoP protocols are designed so that a verifier, with no knowledge of the token before the authentication protocol run, learns nothing about the token from the run. The verifier and CSP may be the same entity, the verifier and relying party may be the same entity or they may all three be separate entities. It is undesirable for verifiers to learn shared secrets unless they are a part of the same entity as the CSP that registered the tokens. Where the verifier and the relying party are separate entities, the verifier must convey the result of the authentication protocol to the relying party. The object created by the verifier to convey this result is called an assertion.[14]
There are four types of authentication schemes: local authentication, centralized authentication, global centralized authentication, global authentication and web application (portal).
When using a local authentication scheme, the application retains the data that pertains to the user's credentials. This information is not usually shared with other applications. The onus is on the user to maintain and remember the types and number of credentials that are associated with the service in which they need to access. This is a high risk scheme because of the possibility that the storage area for passwords might become compromised.
Using the central authentication scheme allows for each user to use the same credentials to access various services. Each application is different and must be designed with interfaces and the ability to interact with a central system to successfully provide authentication for the user. This allows the user to access important information and be able to access private keys that will allow him or her to electronically sign documents.
Using a third party through a global centralized authentication scheme allows the user direct access to authentication services. This then allows the user to access the particular services they need.
The most secure scheme is the global centralized authentication and web application (portal). It is ideal for E-Government use because it allows a wide range of services. It uses a single authentication mechanism involving a minimum of two factors to allow access to required services and the ability to sign documents.[2]
Often, authentication and digital signing are applied in conjunction. Inadvanced electronic signatures, the signatory has authenticated and uniquely linked to a signature. In the case of aqualified electronic signatureas defined in theeIDAS-regulation, the signer's identity is even certified by a qualifiedtrust service provider. This linking of signature and authentication firstly supports the probative value of the signature – commonly referred to asnon-repudiationof origin. The protection of the message on the network-level is called non-repudiation of emission. The authenticated sender and the message content are linked to each other. If a 3rd party tries to change the message content, the signature loses validity.[15]
When developing electronic systems, there are some industry standards requiring United States agencies to ensure the transactions provide an appropriate level of assurance. Generally, servers adopt the US'Office of Management and Budget's (OMB's) E-Authentication Guidance for Federal Agencies (M-04-04) as a guideline, which is published to help federal agencies provide secure electronic services that protect individual privacy. It asks agencies to check whether their transactions require e-authentication, and determine a proper level of assurance.[16]
It established four levels of assurance:[17]
Assurance Level 1: Little or no confidence in the asserted identity's validity.Assurance Level 2: Some confidence in the asserted identity's validity.Assurance Level 3: High confidence in the asserted identity's validity.Assurance Level 4: Very high confidence in the asserted identity's validity.
The OMB proposes a five-step process to determine the appropriate assurance level for their applications:
The required level of authentication assurance are assessed through the factors below:
National Institute of Standards and Technology(NIST) guidance defines technical requirements for each of the four levels of assurance in the following areas:[19]
Triggered by the growth of new cloud solutions and online transactions, person-to-machine and machine-to-machine identities play a significant role in identifying individuals and accessing information. According to the Office of Management and Budget in the U.S., more than $70 million was spent on identity management solutions in both 2013 and 2014.[20]
Governments use e-authentication systems to offer services and reduce time people traveling to a government office. Services ranging from applying for visas to renewing driver's licenses can all be achieved in a more efficient and flexible way. Infrastructure to support e-authentication is regarded as an important component in successful e-government.[21]Poor coordination and poor technical design might be major barriers to electronic authentication.[22]
In several countries there has been established nationwide common e-authentication schemes to ease the reuse of digital identities in different electronic services.[23]Other policy initiatives have included the creation of frameworks for electronic authentication, in order to establish common levels of trust and possibly interoperability between different authentication schemes.[24]
E-authentication is a centerpiece of theUnited States government's effort to expand electronic government, ore-government, as a way of making government more effective and efficient and easier to access. The e-authentication service enables users to access government services online using log-in IDs (identity credentials) from other web sites that both the user and the government trust.
E-authentication is a government-wide partnership that is supported by the agencies that comprise the Federal CIO Council. The United States General Services Administration (GSA) is the lead agency partner. E-authentication works through an association with a trusted credential issuer, making it necessary for the user to log into the issuer's site to obtain the authentication credentials. Those credentials or e-authentication ID are then transferred the supporting government web site causing authentication. The system was created in response a December 16, 2003 memorandum was issued through the Office of Management and Budget. Memorandum M04-04 Whitehouse.[18]That memorandum updates the guidance issued in thePaperwork Elimination Actof 1998, 44 U.S.C. § 3504 and implements section 203 of the E-Government Act, 44 U.S.C. ch. 36.
NIST provides guidelines for digital authentication standards and does away with most knowledge-based authentication methods. A stricter standard has been drafted on more complicated passwords that at least 8 characters long or passphrases that are at least 64 characters long.[25]
InEurope,eIDASprovides guidelines to be used for electronic authentication in regards to electronic signatures and certificate services for website authentication. Once confirmed by the issuing Member State, other participating States are required to accept the user's electronic signature as valid for cross border transactions.
Under eIDAS, electronic identification refers to a material/immaterial unit that contains personal identification data to be used for authentication for an online service. Authentication is referred to as an electronic process that allows for the electronic identification of a natural or legal person. A trust service is an electronic service that is used to create, verify and validate electronic signatures, in addition to creating, verifying and validating certificates for website authentication.
Article 8 of eIDAS allows for the authentication mechanism that is used by a natural or legal person to use electronic identification methods in confirming their identity to a relying party. Annex IV provides requirements for qualified certificates for website authentication.[26][27]
E-authentication is a centerpiece of the Russia government's effort to expand e-government, as a way of making government more effective and efficient and easier for the Russian people to access. The e-authentication service[28]enables users to access government services online using log-in IDs (identity credentials) they already have from web sites that they and the government trust.
Apart from government services, e-authentication is also widely used in other technology and industries. These new applications combine the features of authorizing identities in traditional database and new technology to provide a more secure and diverse use of e-authentication. Some examples are described below.
Mobile authentication is the verification of a user's identity through the use a mobile device. It can be treated as an independent field or it can also be applied with other multifactor authentication schemes in the e-authentication field.[29]
For mobile authentication, there are five levels of application sensitivity from Level 0 to Level 4. Level 0 is for public use over a mobile device and requires no identity authentications, while level 4 has the most multi-procedures to identify users.[30]For either level, mobile authentication is relatively easy to process. Firstly, users send a one-time password (OTP) through offline channels. Then, a server identifies the information and makes adjustment in the database. Since only the user has the access to a PIN code and can send information through their mobile devices, there is a low risk of attacks.[31]
In the early 1980s,electronic data interchange(EDI) systems was implemented, which was considered as an early representative of E-commerce. But ensuring its security is not a significant issue since the systems are all constructed around closed networks. However, more recently, business-to-consumer transactions have transformed. Remote transacting parties have forced the implementation of E-commerce authentication systems.[32]
Generally speaking, the approaches adopted in E-commerce authentication are basically the same as e-authentication. The difference is E-commerce authentication is a more narrow field that focuses on the transactions between customers and suppliers. A simple example of E-commerce authentication includes a client communicating with a merchant server via the Internet. The merchant server usually utilizes a web server to accept client requests, a database management system to manage data and a payment gateway to provideonline paymentservices.[33]
Withself-sovereign identity(SSI) the individual identity holders fully create and control their credentials. Whereas the verifiers can authenticate the provided identities on a decentralized network.
To keep up with the evolution of services in the digital world, there is continued need for security mechanisms. While passwords will continue to be used, it is important to rely on authentication mechanisms, most importantly multifactor authentication. As the usage of e-signatures continues to significantly expand throughout the United States, the EU and throughout the world, there is expectation that regulations such aseIDASwill eventually be amended to reflect changing conditions along with regulations in the United States.[34]
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Intelecommunications,out-of-bandactivity is activity outside a definedfrequency band, or, metaphorically, outside of any primarycommunication channel. Protection fromfalsingis among its purposes.
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Reliance authenticationis a part of the trust-based identity attribution process whereby a second entity relies upon theauthenticationprocesses put in place by a first entity. The second entity creates a further element that is unique and specific to its purpose, that can only be retrieved or accessed by the authentication processes of the first entity having first being met.
Reliance authentication can be achieved by one or more tokens with random characteristics being transmitted to a secure area controlled by the first entity, where such secure area is only accessible by the person authorised to use the account. The secure area may be an online banking portal, telephone banking system, or mobile banking application.
The token is often in the form of a single or plural of debit or credits to a financial account, where the numerical values of the debit or credits form the token, whose numeric value is to be confirmed by the account holder.
The token are retrieved by the cardholder accessing a secure area from the first entity's secure area, which is protected and accessible only by satisfying the first entity's authentication means. In the case of financial services, authentication to access the secure area normally includesmulti-factorand in theSEPAwould likely involve strong authentication.
The transmission and requirement to retrieve the token adds a furtherchallenge and responsefactor to the overall authentication process when considered from the point of view of the second party, which generates and transmits the token.
The token may be generated by the second party dynamically, and can thus act as aone-time password.
The reliance authentication method has particular application with financial instruments such ascredit cards,e-mandateanddirect debittransactions, whereby a person may instigate a transaction on a financial instrument, however the financial instrument is not verified as belonging to that person until that person confirms the value of the token.
The reliance method often incorporates anout-of-bandresponse means, once the tokens have been retrieved from the secure area.
Reliance authentication uses multi-step inputs to ensure that the user is not a fraud. Some examples include:
The introduction of strong customer authentication[2]for online payment transactions within the European Union now links a verified person to an account, where such person has been identified in accordance with statutory requirements prior to the account being opened. Reliance authentication makes use of pre-existing accounts, to piggyback further services upon those accounts, providing that the original source is 'reliable'.
The concept of reliability is a legal one derived from various anti money laundering (AML) / counter-terrorism funding (CTF) legislation in the USA,[3]EU28,[4]Australia,[5]Singapore and New Zealand[6]where second parties may place reliance on the customer due diligence process of the first party, where the first party is say a financial institution.
In the Australian legislation, 'reliance' is based upon section 38 of theAnti-Money Laundering and Counter-Terrorism Financing Act 2006(Cth).
In the European Commission'sProposal for a Directive of the European Parliament and of the Council on the prevention of the use of the financial system for the purpose of money laundering and terrorist financing, reliance is based upon Article 11(1)(a).
Reliance in the UK has a very specific meaning and relates to the process under Regulation 17 of theMoney Laundering Regulations 2007. "Reliance" for the purpose of AML and "reliance authentication" are not the same, although both use similar concepts.
The Federal Financial Institutions Examination Council of the United States of America (FFIEC) issued"Authentication in an Internet Banking Environment", dated October 2005. Reliance authentication is outlined per the final paragraph of page 14.
Advantages of reliance authentication methods are:
Disadvantages of reliance authentication methods are:
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Universal 2nd Factor(U2F) is an open standard that strengthens and simplifiestwo-factor authentication(2FA) using specializedUniversal Serial Bus(USB),near-field communication(NFC), orBluetooth Low Energy(BLE) devices based on similar security technology found insmart cards.[1][2][3][4][5]It is succeeded by theFIDO2 Project, which includes theW3CWeb Authentication (WebAuthn) standard and theFIDO Alliance'sClient to Authenticator Protocol2 (CTAP2).[6]
While initially developed byGoogleandYubico, with contribution fromNXP Semiconductors, the standard is now hosted by the FIDO Alliance.[7][8]
Whiletime-based one-time password(TOTPs) (e.g. 6-digit codes generated on Google Authenticator) were a significant improvement over SMS-based security codes, a number of security vulnerabilities were still possible to exploit, which U2F sought to improve. Specifically:
In terms of disadvantages, one significant difference and potential drawback to be considered regarding hardware-based U2F solutions is that unlike with TOTP shared-secret methods, there is no possibility of "backing up" recovery codes or shared secrets. If a hardware duplicate or alternative hardware key is not kept and the original U2F hardware key is lost, no recovery of the key is possible (because the private key exists only in hardware). Therefore, for services that do not provide any alternative account recovery method, the use of U2F should be carefully considered.
The USB devices communicate with the host computer using thehuman interface device(HID) protocol, essentially mimicking a keyboard.[9][failed verification–see discussion]This avoids the need for the user to install special hardware driver software in the host computer and permits application software (such as a browser) to directly access the security features of the device without user effort other than possessing and inserting the device. Once communication is established, the application exercises achallenge–response authenticationwith the device usingpublic-key cryptographymethods and a secret unique device key manufactured into the device.[10]
The device key is vulnerable tomalicious manufacturer duplication.[11]
In 2020, independent security researchers found a method to extract private keys from Google Titan Key, a popular U2F hardware security token.[12][13][14]The method required physical access to the key for several hours, several thousand euros-worth of equipment, and was destructive to the plastic case of the key.[12][13][14]The attackers concluded that the difficulty of the attack meant that people were still safer to use the keys than not.[12][13][14]The attack was possible due to a vulnerability in the A700X microchip made byNXP Semiconductors, which is also used in security tokens made byFeitianandYubico, meaning that those tokens are also vulnerable.[12][15]The vulnerability wasresponsibly disclosedto the affected manufacturers so that it might be fixed in future products.[12][13][14]
U2F security keys are supported byGoogle Chromesince version 38,[2]Firefoxsince version 57[16]andOperasince version 40. U2F security keys can be used as an additional method of two-step verification on online services that support the U2F protocol, including Google,[2]Azure,[17]Dropbox,[18]GitHub,[19]GitLab,[20]Bitbucket,[21]Nextcloud,[22]Facebook,[23]and others.[24]
Chrome,Firefox, and Opera were, as of 2015[update], the only browsers supporting U2F natively.Microsofthas enabled FIDO 2.0 support forWindows 10'sWindows Hellologin platform.[25]MicrosoftEdge[26]browser gained support for U2F in the October 2018 Windows Update.Microsoft accounts, includingOffice 365,OneDrive, and other Microsoft services, do not yet have U2F support.Mozillahas integrated it into Firefox 57, and enabled it by default in Firefox 60[27][28][29][30]andThunderbird60.[31]Microsoft Edge starting from build 17723 support FIDO2.[32]As of iOS and iPadOS 13.3 Apple now supports U2F in the Safari browser on those platforms.
The U2F standard has undergone two major revisions:
Additional specification documents may be obtained from the FIDO web site.[35]
The U2F 1.0 Proposed Standard (October 9, 2014) was the starting point for a short-lived specification known as the FIDO 2.0 Proposed Standard (September 4, 2015). The latter was formally submitted to theWorld Wide Web Consortium(W3C) on November 12, 2015.[36]Subsequently, the first Working Draft of the W3C Web Authentication (WebAuthn) standard was published on May 31, 2016. The WebAuthn standard has been revised numerous times since then, becoming a W3C Recommendation on March 4, 2019.
Meanwhile the U2F 1.2 Proposed Standard (April 11, 2017) became the starting point for theClient to Authenticator Protocol(CTAP) Proposed Standard, which was published on September 27, 2017. FIDO CTAP complements W3C WebAuthn, both of which are in scope for theFIDO2 Project.
WebAuthn and CTAP provide a complete replacement for U2F, which has been renamed "CTAP1" in the latest version of the FIDO2 standard.[37]The WebAuthn protocol is backward-compatible (via the AppID extension) with U2F-onlysecurity keys[38]but the U2F protocol is not compatible with a WebAuthn-onlyauthenticator.[39][37]Some authenticators support both U2F and WebAuthn while some WebAuthn clients support keys created via the legacy U2F API.[citation needed]
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Identity threat detection and response (ITDR)is acybersecuritydiscipline that includes tools and best practices to protectidentity managementinfrastructure from attacks. ITDR can block and detectthreats, verify administrator credentials, respond to various attacks, and restore normal operations.[1]Common identity threats includephishing, stolen credentials,insider threats, andransomware.[2]
ITDR adds an extra layer of security toidentity and access management(IAM) systems. It helps secure accounts, permissions, and the identity infrastructure itself from compromise. With attackers targeting identity tools directly, ITDR is becoming more important in 2023 : according toGartner, established IAM hygiene practices likeprivileged access managementand identity governance are no longer enough.[1]
ITDR can be part of azero trust security model. ITDR is especially relevant formulticloudinfrastructures, which have gaps between cloud providers' distinct IAM implementations. Closing these gaps and orchestrating identity across clouds is an ITDR focus.[3]
ITDR enhancesidentity and access management(IAM) by adding detection and response capabilities. It provides visibility into potential credential misuse and abuse of privileges. ITDR also finds gaps left by IAM andprivileged access management(PAM) systems.[4]ITDR requires monitoring identity systems for misuse and compromise. It uses lower latency detections than general security systems. ITDR involves coordination between IAM and security teams.[1]
ITDR uses theMITRE ATT&CKframework against known attack vectors. It combines foundational IAM controls likemulti-factor authenticationwith monitoring. ITDR prevents compromise of admin accounts and credentials. It modernizes infrastructure through standards likeOAuth 2.0.
Organizations adopt ITDR to complement IAM andendpoint detection and response. ITDR specifically monitors identity systems and user activity logs for attacks. It can isolate affected systems and gather forensic data. Adoption requires budget, training, and buy-in. Organizations can start with IAM fundamentals likemulti-factor authenticationandrole-based access control.[4]
ITDR tools can find misconfigurations inActive Directory. Strategies can update firewalls, intrusion systems, and security apps. ITDR integrates withSIEMtools for threat monitoring and automated response. An ITDR incident response plan handles compromised credentials and privilege escalation. Awareness training teaches users to spot identity-based attacks.[4]
ITDR emerged as a distinct cybersecurity segment in 2022. The term was coined byGartner.[4]
According toGartner, ITDR vendors include Authomize,CrowdStrike, Gurucul,Microsoft, Netwrix, Oort,Proofpoint, Quest Software, Semperis,SentinelOne, and Silverfort.[1]
While EDR detects issues on endpoints, ITDR concentrates on monitoring and analyzing user activity and access management logs to uncover malicious activity. It gathers data from multiple identity and access management (IAM) sources across on-premises and cloud environments. Together they give a more complete picture to improve detection and response to sophisticated attacks involving lateral movement and identity deception.[5]
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Abstract algebrais the subject area ofmathematicsthat studiesalgebraic structures, such asgroups,rings,fields,modules,vector spaces, andalgebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns andrealorcomplex numbers, often now calledelementary algebra. The distinction is rarely made in more recent writings.
Algebraic structuresare defined primarily assetswithoperations.
Structure preserving maps calledhomomorphismsare vital in the study of algebraic objects.
There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra.
Advanced concepts:
Representation theory
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The followingoutlineis provided as an overview of and guide tocategory theory, the area of study inmathematicsthat examines in anabstractway the properties of particular mathematical concepts, by formalising them as collections ofobjectsandarrows(also calledmorphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
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This is alist ofLie grouptopics, by Wikipedia page.
SeeTable of Lie groupsfor a list
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Geometric group theoryis an area inmathematicsdevoted to the study offinitely generated groupsvia exploring the connections betweenalgebraicproperties of suchgroupsandtopologicalandgeometricproperties of spaces on which these groups canactnon-trivially (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying theCayley graphsof groups, which, in addition to thegraphstructure, are endowed with the structure of ametric space, given by the so-calledword metric.
Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts withlow-dimensional topology,hyperbolic geometry,algebraic topology,computational group theoryanddifferential geometry. There are also substantial connections withcomplexity theory,mathematical logic, the study ofLie groupsand their discrete subgroups,dynamical systems,probability theory,K-theory, and other areas of mathematics.
In the introduction to his bookTopics in Geometric Group Theory,Pierre de la Harpewrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, and reminds me of several things thatGeorges de Rhampracticed on many occasions, such as teaching mathematics, recitingMallarmé, or greeting a friend".[1]: 3
Geometric group theory grew out ofcombinatorial group theorythat largely studied properties ofdiscrete groupsvia analyzinggroup presentations, which describe groups asquotientsoffree groups; this field was first systematically studied byWalther von Dyck, student ofFelix Klein, in the early 1880s,[2]while an early form is found in the 1856icosian calculusofWilliam Rowan Hamilton, where he studied theicosahedral symmetrygroup via the edge graph of thedodecahedron. Currently combinatorial group theory as an area is largely subsumed by geometric group theory. Moreover, the term "geometric group theory" came to often include studying discrete groups using probabilistic,measure-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal.
In the first half of the 20th century, pioneering work ofMax Dehn,Jakob Nielsen,Kurt ReidemeisterandOtto Schreier,J. H. C. Whitehead,Egbert van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups.[3]Other precursors of geometric group theory includesmall cancellation theoryandBass–Serre theory. Small cancellation theory was introduced byMartin Grindlingerin the 1960s[4][5]and further developed byRoger LyndonandPaul Schupp.[6]It studiesvan Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions and derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre,[7]derives structural algebraic information about groups by studying group actions onsimplicial trees.
External precursors of geometric group theory include the study of lattices in Lie groups, especiallyMostow's rigidity theorem, the study ofKleinian groups, and the progress achieved inlow-dimensional topologyand hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, byWilliam Thurston'sGeometrization program.
The emergence of geometric group theory as a distinct area of mathematics is usually traced to the late 1980s and early 1990s. It was spurred by the 1987 monograph ofMikhail Gromov"Hyperbolic groups"[8]that introduced the notion of ahyperbolic group(also known asword-hyperbolicorGromov-hyperbolicornegatively curvedgroup), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monographAsymptotic Invariants of Infinite Groups,[9]that outlined Gromov's program of understanding discrete groups up toquasi-isometry. The work of Gromov had a transformative effect on the study of discrete groups[10][11][12]and the phrase "geometric group theory" started appearing soon afterwards. (see e.g.[13]).
Notable themes and developments in geometric group theory in 1990s and 2000s include:
The following examples are often studied in geometric group theory:
These texts cover geometric group theory and related topics.
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Inmathematics, asystem of parametersfor alocalNoetherian ringofKrull dimensiondwithmaximal idealmis a set of elementsx1, ...,xdthat satisfies any of the following equivalent conditions:
Every local Noetherian ring admits a system of parameters.[1]
It is not possible for fewer thandelements to generate an ideal whose radical ismbecause then the dimension ofRwould be less thand.
IfMis ak-dimensional module over a local ring, thenx1, ...,xkis asystem of parametersforMif thelengthofM/ (x1, ...,xk)Mis finite.
Thiscommutative algebra-related article is astub. You can help Wikipedia byexpanding it.
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Inmathematics, afilterororder filteris a specialsubsetof apartially ordered set(poset), describing "large" or "eventual" elements. Filters appear inorderandlattice theory, but alsotopology, whence they originate. The notiondualto a filter is anorder ideal.
Special cases of filters includeultrafilters, which are filters that cannot be enlarged, and describe nonconstructive techniques inmathematical logic.
Filters on setswere introduced byHenri Cartanin 1937.Nicolas Bourbaki, in their bookTopologie Générale, popularized filters as an alternative toE. H. MooreandHerman L. Smith's 1922 notion of anet; order filters generalize this notion from the specific case of apower setunderinclusionto arbitrarypartially ordered sets. Nevertheless, thetheory of power-set filtersretains interest in its own right, in part for substantialapplications in topology.
Fix apartially ordered set (poset)P. Intuitively, a filterFis a subset ofPwhose members are elements large enough to satisfy some criterion.[1]For instance, ifx∈P, then the set of elements abovexis a filter, called the principal filter atx. (Ifxandyareincomparableelements ofP, then neither the principal filter atxnoryis contained in the other.)
Similarly, a filter on a setScontains those subsets that are sufficiently large to contain some giventhing. For example, ifSis thereal lineandx∈S, then the family of sets includingxin theirinterioris a filter, called the neighborhood filter atx. Thethingin this case is slightly larger thanx, but it still does not contain any other specific point of the line.
The above considerations motivate the upward closure requirement in thedefinition below: "large enough" objects can always be made larger.
To understand the other two conditions, reverse the roles and instead considerFas a "locating scheme" to findx. In this interpretation, one searches in some spaceX, and expectsFto describe those subsets ofXthat contain the goal. The goal must be located somewhere; thus theempty set∅can never be inF. And if two subsets both contain the goal, then should "zoom in" to their common region.
An ultrafilter describes a "perfect locating scheme" where each scheme component gives new information (either "look here" or "look elsewhere").Compactnessis the property that "every search is fruitful," or, to put it another way, "every locating scheme ends in a search result."
A common use for a filter is to define properties that are satisfied by "generic" elements of some topological space.[2]This application generalizes the "locating scheme" to find points that might be hard to write down explicitly.
A subsetFof a partially ordered set(P, ≤)is afilterordual idealif the following are satisfied:[3]
If, additionally,F≠P, thenFis said to be aproper filter. Authors inset theoryandmathematical logicoften require all filters to be proper;[4]this article willeschewthat convention. Anultrafilteris a proper filter contained in no other proper filter except itself.
A subsetSofFis abaseorbasisforFif theupper setgenerated byS(i.e., the smallest upwards-closed set containingS) is equal toF. Since every filter is upwards-closed, every filter is a base for itself.
Moreover, ifB⊆Pis nonempty and downward directed, thenBgenerates an upper setFthat is a filter (for whichBis a base). Such sets are calledprefilters, as well as the aforementionedfilter base/basis, andFis said to begeneratedorspannedbyB. A prefilter is proper if and only if it generates a proper filter.
Givenp∈P, the set{x:p≤x}is the smallest filter containingp, and sometimes written↑p. Such a filter is called aprincipal filter;pis said to be theprincipal elementofF, or generateF.
SupposeBandCare two prefilters onP, and, for eachc∈C, there is ab∈B, such thatb≤c. Then we say thatBisfinerthan (orrefines)C; likewise,Ciscoarserthan (orcoarsens)B. Refinement is apreorderon the set of prefilters. In fact, ifCalso refinesB, thenBandCare calledequivalent, for they generate the same filter. Thus passage from prefilter to filter is an instance of passing from a preordering to associated partial ordering.
Historically, filters generalized toorder-theoretic latticesbefore arbitrary partial orders. In the case of lattices, downward direction can be written as closure under finitemeets: for allx,y∈F, one hasx∧y∈F.[3]
A linear (ultra)filter is an (ultra)filter on thelatticeofvector subspacesof a givenvector space, ordered by inclusion. Explicitly, a linear filter on a vector spaceXis a familyBof vector subspaces ofXsuch that ifA,B∈BandCis a vector subspace ofXthat containsA, thenA∩B∈BandC∈B.[5]
A linear filter is proper if it does not contain{0}.[5]
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
Given a setS, thepower setP(S)ispartially orderedbyset inclusion; filters on this poset are often just called "filters onS," in anabuse of terminology. For such posets, downward direction and upward closure reduce to:[4]
Aproper[7]/non-degenerate[8]filter is one that does not contain∅, and these three conditions (including non-degeneracy) areHenri Cartan's original definition of a filter.[9][10]It is common —though not universal— to require filters on sets to be proper (whatever one's stance on poset filters); we shall again eschew this convention.
Prefilters on a set are proper if and only if they do not contain∅either.
For every subsetTofP(S), there is a smallest filterFcontainingT. As with prefilters,Tis said to generate or spanF; a base forFis the setUof all finite intersections ofT. The setTis said to be afilter subbasewhenF(and thusU) is proper.
Proper filters on sets have thefinite intersection property.
IfS= ∅, thenSadmits only the improper filter{∅}.
A filter is said to be afree filterif the intersection of its members is empty. A proper principal filter is not free.
Since the intersection of any finite number of members of a filter is also a member, no proper filter on a finite set is free, and indeed is the principal filter generated by the common intersection of all of its members. But a nonprincipal filter on an infinite set is not necessarily free: a filter is free if and only if it includes theFréchet filter(see§ Examples).
See the image at the top of this article for a simple example of filters on the finite posetP({1, 2, 3, 4}).
Partially orderℝ → ℝ, the space of real-valued functions onℝ, by pointwise comparison. Then the set of functions "large at infinity,"{f:limx→±∞f(x)=∞},{\displaystyle \left\{f:\lim _{x\to \pm \infty }{f(x)}=\infty \right\}{\text{,}}}is a filter onℝ → ℝ. One can generalize this construction quite far bycompactifyingthe domain andcompletingthe codomain: ifXis a set with distinguished subsetSandYis a poset with distinguished elementm, then{f:f|S≥m}is a filter inX→Y.
The set{{k:k≥N} :N∈ ℕ}is a filter inP(ℕ). More generally, ifDis anydirected set, then{{k:k≥N}:N∈D}{\displaystyle \{\{k:k\geq N\}:N\in D\}}is a filter inP(D), called the tail filter. Likewise anynet{xα}α∈Αgenerates the eventuality filter{{xβ: α ≤ β} : α ∈ Α}. A tail filter is the eventuality filter forxα= α.
TheFréchet filteron an infinite setXis{A:X∖Afinite}.{\displaystyle \{A:X\setminus A{\text{ finite}}\}{\text{.}}}If(X, μ)is ameasure space, then the collection{A: μ(X∖A) = 0}is a filter. Ifμ(X) = ∞, then{A: μ(X∖A) < ∞}is also a filter; the Fréchet filter is the case whereμiscounting measure.
Given an ordinala, a subset ofais called aclubif it is closed in theorder topologyofabut has net-theoretic limita. The clubs ofaform a filter: theclub filter,♣(a).
The previous construction generalizes as follows: any clubCis also a collection of dense subsets (in theordinal topology) ofa, and♣(a)meets each element ofC. ReplacingCwith an arbitrary collectionC̃ofdense sets, there "typically" exists a filter meeting each element ofC̃, called ageneric filter. For countableC̃, theRasiowa–Sikorski lemmaimplies that such a filter must exist; for "small"uncountableC̃, the existence of such a filter can beforcedthroughMartin's axiom.
LetPdenote the set ofpartial ordersoflimited cardinality,moduloisomorphism. Partially orderPby:
Then the subset ofnon-atomicpartial orders forms a filter. Likewise, ifIis the set ofinjective modulesover some givencommutative ring, of limited cardinality, modulo isomorphism, then a partial order onIis:
Given any infinite cardinalκ, the modules inIthat cannot be generated by fewer thanκelements form a filter.
Everyuniform structureon a setXis a filter onX×X.
Thedual notionto a filter — that is, the concept obtained by reversing all≤and exchanging∧with∨— is an order ideal. Because of this duality, any question of filters can be mechanically translated to a question about ideals and vice versa; in particular, aprimeormaximalfilter is a filter whose corresponding ideal is (respectively) prime or maximal.
A filter is an ultrafilter if and only if the corresponding ideal is minimal.
For every filterFon a setS, the set function defined bym(A)={1ifA∈F0ifS∖A∈Fis undefinedotherwise{\displaystyle m(A)={\begin{cases}1&{\text{if }}A\in F\\0&{\text{if }}S\smallsetminus A\in F\\{\text{is undefined}}&{\text{otherwise}}\end{cases}}}is finitely additive — a "measure," if that term is construed rather loosely. Moreover, the measures so constructed are defined everywhere ifFis anultrafilter. Therefore, the statement{x∈S:φ(x)}∈F{\displaystyle \left\{\,x\in S:\varphi (x)\,\right\}\in F}can be considered somewhat analogous to the statement thatφholds "almost everywhere." That interpretation of membership in a filter is used (for motivation, not actualproofs) in the theory ofultraproductsinmodel theory, a branch ofmathematical logic.
Ingeneral topologyand analysis, filters are used to define convergence in a manner similar to the role ofsequencesin ametric space. They unify the concept of alimitacross the wide variety of arbitrarytopological spaces.
To understand the need for filters, begin with the equivalent concept of anet. Asequenceis usually indexed by thenatural numbersℕ, which are atotally ordered set. Nets generalize the notion of a sequence by replacingℕwith an arbitrarydirected set. In certain categories of topological spaces, such asfirst-countable spaces, sequences characterize most topological properties, but this is not true in general. However, nets — as well as filters — always do characterize those topological properties.
Filters do not involve any set external to the topological spaceX, whereas sequences and nets rely on other directed sets. For this reason, the collection of all filters onXis always aset, whereas the collection of allX-valued nets is aproper class.
Any pointxin the topological spaceXdefines aneighborhood filter or systemNx: namely, the family of all sets containingxin theirinterior. A setNof neighborhoods ofxis aneighborhood baseatxifNgeneratesNx. Equivalently,S⊆Xis a neighborhood ofxif and only if there existsN∈Nsuch thatN⊆S.
A prefilterBconvergesto a pointx, writtenB→x, if and only ifBgenerates a filterFthat contains the neighborhood filterNx— explicitly, for every neighborhoodUofx, there is someV∈Bsuch thatV⊆U. Less explicitly,B→xif and only ifBrefinesNx, and any neighborhood base atxcan replaceNxin this condition. Clearly, everyneighborhood baseatxconverges tox.
A filterF(which generates itself) converges toxifNx⊆F. The above can also be reversed to characterize the neighborhood filterNx:Nxis the finest filter coarser than each filter converging tox.
IfB→x, thenxis called alimit(point) ofB. The prefilterBis said to cluster atx(or havexas acluster point) if and only if each element ofBhas non-empty intersection with each neighborhood ofx. Every limit point is a cluster point but the converse is not true in general. However, every cluster point of anultrafilter is a limit point.
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In the mathematical field ofset theory, anidealis apartially orderedcollection ofsetsthat are considered to be "small" or "negligible". Everysubsetof an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and theunionof any two elements of the ideal must also be in the ideal.
More formally, given a setX,{\displaystyle X,}an idealI{\displaystyle I}onX{\displaystyle X}is anonemptysubset of thepowersetofX,{\displaystyle X,}such that:
Some authors add a fourth condition thatX{\displaystyle X}itself is not inI{\displaystyle I}; ideals with this extra property are calledproper ideals.
Ideals in the set-theoretic sense are exactlyideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exactlyideals in the ring-theoretic senseon theBoolean ringformed by the powerset of the underlying set. The dual notion of an ideal is afilter.
An element of an idealI{\displaystyle I}is said to beI{\displaystyle I}-nullorI{\displaystyle I}-negligible, or simplynullornegligibleif the idealI{\displaystyle I}is understood from context. IfI{\displaystyle I}is an ideal onX,{\displaystyle X,}then a subset ofX{\displaystyle X}is said to beI{\displaystyle I}-positive(or justpositive) if it isnotan element ofI.{\displaystyle I.}The collection of allI{\displaystyle I}-positive subsets ofX{\displaystyle X}is denotedI+.{\displaystyle I^{+}.}
IfI{\displaystyle I}is a proper ideal onX{\displaystyle X}and for everyA⊆X{\displaystyle A\subseteq X}eitherA∈I{\displaystyle A\in I}orX∖A∈I,{\displaystyle X\setminus A\in I,}thenI{\displaystyle I}is aprime ideal.
Given idealsIandJon underlying setsXandYrespectively, one forms theskeworFubiniproductI×J{\displaystyle I\times J}, an ideal on theCartesian productX×Y,{\displaystyle X\times Y,}as follows: For any subsetA⊆X×Y,{\displaystyle A\subseteq X\times Y,}A∈I×Jif and only if{x∈X:{y:⟨x,y⟩∈A}∉J}∈I{\displaystyle A\in I\times J\quad {\text{ if and only if }}\quad \{x\in X\;:\;\{y:\langle x,y\rangle \in A\}\not \in J\}\in I}That is, a set lies in the product ideal if only a negligible collection ofx-coordinates correspond to a non-negligible slice ofAin they-direction. (Perhaps clearer: A set ispositivein the product ideal if positively manyx-coordinates correspond to positive slices.)
An idealIon a setXinduces anequivalence relationon℘(X),{\displaystyle \wp (X),}the powerset ofX, consideringAandBto be equivalent (forA,B{\displaystyle A,B}subsets ofX) if and only if thesymmetric differenceofAandBis an element ofI. Thequotientof℘(X){\displaystyle \wp (X)}by this equivalence relation is aBoolean algebra, denoted℘(X)/I{\displaystyle \wp (X)/I}(read "P ofXmodI").
To every ideal there is a correspondingfilter, called itsdual filter. IfIis an ideal onX, then the dual filter ofIis the collection of all setsX∖A,{\displaystyle X\setminus A,}whereAis an element ofI. (HereX∖A{\displaystyle X\setminus A}denotes therelative complementofAinX; that is, the collection of all elements ofXthat arenotinA).
IfI{\displaystyle I}andJ{\displaystyle J}are ideals onX{\displaystyle X}andY{\displaystyle Y}respectively,I{\displaystyle I}andJ{\displaystyle J}areRudin–Keisler isomorphicif they are the same ideal except for renaming of the elements of their underlying sets (ignoring negligible sets). More formally, the requirement is that there be setsA{\displaystyle A}andB,{\displaystyle B,}elements ofI{\displaystyle I}andJ{\displaystyle J}respectively, and abijectionφ:X∖A→Y∖B,{\displaystyle \varphi :X\setminus A\to Y\setminus B,}such that for any subsetC⊆X,{\displaystyle C\subseteq X,}C∈I{\displaystyle C\in I}if and only if theimageofC{\displaystyle C}underφ∈J.{\displaystyle \varphi \in J.}
IfI{\displaystyle I}andJ{\displaystyle J}are Rudin–Keisler isomorphic, then℘(X)/I{\displaystyle \wp (X)/I}and℘(Y)/J{\displaystyle \wp (Y)/J}are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are calledtrivial isomorphisms.
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In mathematics, asemigroupis analgebraic structureconsisting of asettogether with anassociativeinternalbinary operationon it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmeticmultiplication):x⋅y, or simplyxy, denotes the result of applying the semigroup operation to theordered pair(x,y). Associativity is formally expressed as that(x⋅y) ⋅z=x⋅ (y⋅z)for allx,yandzin the semigroup.
Semigroups may be considered a special case ofmagmas, where the operation is associative, or as a generalization ofgroups, without requiring the existence of an identity element or inverses.[a]As in the case of groups or magmas, the semigroup operation need not becommutative, sox⋅yis not necessarily equal toy⋅x; a well-known example of an operation that is associative but non-commutative ismatrix multiplication. If the semigroup operation is commutative, then the semigroup is called acommutative semigroupor (less often than in theanalogous case of groups) it may be called anabelian semigroup.
Amonoidis an algebraic structure intermediate between semigroups and groups, and is a semigroup having anidentity element, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid. A natural example isstringswithconcatenationas the binary operation, and the empty string as the identity element. Restricting to non-emptystringsgives an example of a semigroup that is not a monoid. Positiveintegerswith addition form a commutative semigroup that is not a monoid, whereas the non-negativeintegersdo form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused withquasigroups, which are generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroupspreserve from groupsthe notion ofdivision. Division in semigroups (or in monoids) is not possible in general.
The formal study of semigroups began in the early 20th century. Early results includea Cayley theorem for semigroupsrealizing any semigroup as atransformation semigroup, in which arbitrary functions replace the role of bijections in group theory. A deep result in the classification of finite semigroups isKrohn–Rhodes theory, analogous to theJordan–Hölder decompositionfor finite groups. Some other techniques for studying semigroups, likeGreen's relations, do not resemble anything in group theory.
The theory of finite semigroups has been of particular importance intheoretical computer sciencesince the 1950s because of the natural link between finite semigroups andfinite automatavia thesyntactic monoid. Inprobability theory, semigroups are associated withMarkov processes.[1]In other areas ofapplied mathematics, semigroups are fundamental models forlinear time-invariant systems. Inpartial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time.
There are numerousspecial classes of semigroups, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention:regular semigroups,orthodox semigroups,semigroups with involution,inverse semigroupsandcancellative semigroups. There are also interesting classes of semigroups that do not contain any groups except thetrivial group; examples of the latter kind arebandsand their commutative subclass –semilattices, which are alsoordered algebraic structures.
A semigroup is asetStogether with abinary operation⋅ (that is, afunction⋅ :S×S→S) that satisfies theassociative property:
More succinctly, a semigroup is an associativemagma.
Aleft identityof a semigroupS(or more generally,magma) is an elementesuch that for allxinS,e⋅x=x. Similarly, aright identityis an elementfsuch that for allxinS,x⋅f=x. Left and right identities are both calledone-sided identities. A semigroup may have one or more left identities but no right identity, and vice versa.
Atwo-sided identity(or justidentity) is an element that is both a left and right identity. Semigroups with a two-sided identity are calledmonoids. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup. If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore the unique one-sided identity).
A semigroupSwithout identity may beembeddedin a monoid formed by adjoining an elemente∉StoSand defininge⋅s=s⋅e=sfor alls∈S∪ {e}.[2][3]The notationS1denotes a monoid obtained fromSby adjoining an identityif necessary(S1=Sfor a monoid).[3]
Similarly, every magma has at most oneabsorbing element, which in semigroup theory is called azero. Analogous to the above construction, for every semigroupS, one can defineS0, a semigroup with 0 that embedsS.
The semigroup operation induces an operation on the collection of its subsets: given subsetsAandBof a semigroupS, their productA·B, written commonly asAB, is the set{ab|a∈Aandb∈B}.(This notion is defined identically asit is for groups.) In terms of this operation, a subsetAis called
IfAis both a left ideal and a right ideal then it is called anideal(or atwo-sided ideal).
IfSis a semigroup, then the intersection of any collection of subsemigroups ofSis also a subsemigroup ofS.
So the subsemigroups ofSform acomplete lattice.
An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of acommutativesemigroup, when it exists, is a group.
Green's relations, a set of fiveequivalence relationsthat characterise the elements in terms of theprincipal idealsthey generate, are important tools for analysing the ideals of a semigroup and related notions of structure.
The subset with the property that every element commutes with any other element of the semigroup is called thecenterof the semigroup.[4]The center of a semigroup is actually a subsemigroup.[5]
Asemigrouphomomorphismis a function that preserves semigroup structure. A functionf:S→Tbetween two semigroups is a homomorphism if the equation
holds for all elementsa,binS, i.e. the result is the same when performing the semigroup operation after or before applying the mapf.
A semigroup homomorphism between monoids preserves identity if it is amonoid homomorphism. But there are semigroup homomorphisms that are not monoid homomorphisms, e.g. the canonical embedding of a semigroupSwithout identity intoS1. Conditions characterizing monoid homomorphisms are discussed further. Letf:S0→S1be a semigroup homomorphism. The image offis also a semigroup. IfS0is a monoid with an identity elemente0, thenf(e0) is the identity element in the image off. IfS1is also a monoid with an identity elemente1ande1belongs to the image off, thenf(e0) =e1, i.e.fis a monoid homomorphism. Particularly, iffissurjective, then it is a monoid homomorphism.
Two semigroupsSandTare said to beisomorphicif there exists abijectivesemigroup homomorphismf:S→T. Isomorphic semigroups have the same structure.
Asemigroup congruence~ is anequivalence relationthat is compatible with the semigroup operation. That is, a subset~ ⊆S×Sthat is an equivalence relation andx~yandu~vimpliesxu~yvfor everyx,y,u,vinS. Like any equivalence relation, a semigroup congruence ~ inducescongruence classes
and the semigroup operation induces a binary operation ∘ on the congruence classes:
Because ~ is a congruence, the set of all congruence classes of ~ forms a semigroup with ∘, called thequotient semigrouporfactor semigroup, and denotedS/ ~. The mappingx↦ [x]~is a semigroup homomorphism, called thequotient map,canonicalsurjectionorprojection; ifSis a monoid then quotient semigroup is a monoid with identity [1]~. Conversely, thekernelof any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of thefirst isomorphism theorem in universal algebra. Congruence classes and factor monoids are the objects of study instring rewriting systems.
Anuclear congruenceonSis one that is the kernel of an endomorphism ofS.[6]
A semigroupSsatisfies themaximal condition on congruencesif any family of congruences onS, ordered by inclusion, has a maximal element. ByZorn's lemma, this is equivalent to saying that theascending chain conditionholds: there is no infinite strictly ascending chain of congruences onS.[7]
Every idealIof a semigroup induces a factor semigroup, theRees factor semigroup, via the congruence ρ defined byxρyif eitherx=y, or bothxandyare inI.
The following notions[8]introduce the idea that a semigroup is contained in another one.
A semigroupTis a quotient of a semigroupSif there is a surjective semigroup morphism fromStoT. For example,(Z/2Z, +)is a quotient of(Z/4Z, +), using the morphism consisting of taking the remainder modulo 2 of an integer.
A semigroupTdivides a semigroupS, denotedT≼SifTis a quotient of a subsemigroupS. In particular, subsemigroups ofSdividesT, while it is not necessarily the case that there are a quotient ofS.
Both of those relations are transitive.
For any subsetAofSthere is a smallest subsemigroupTofSthat containsA, and we say thatAgeneratesT. A single elementxofSgenerates the subsemigroup{xn|n∈Z+}. If this is finite, thenxis said to be offinite order, otherwise it is ofinfinite order.
A semigroup is said to beperiodicif all of its elements are of finite order.
A semigroup generated by a single element is said to bemonogenic(orcyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positiveintegerswith the operation of addition.
If it is finite and nonempty, then it must contain at least oneidempotent.
It follows that every nonempty periodic semigroup has at least one idempotent.
A subsemigroup that is also a group is called asubgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotenteof the semigroup there is a unique maximal subgroup containinge. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the termmaximal subgroupdiffers from its standard use in group theory.
More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimalidealand at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for a set of two elements{a,b}, eight form semigroups[b]whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, seeKrohn–Rhodes theory.
There is a structure theorem for commutative semigroups in terms ofsemilattices.[10]A semilattice (or more precisely a meet-semilattice)(L, ≤)is apartially ordered setwhere every pair of elementsa,b∈Lhas agreatest lower bound, denoteda∧b. The operation ∧ makesLinto a semigroup that satisfies the additionalidempotencelawa∧a=a.
Given a homomorphismf:S→Lfrom an arbitrary semigroup to a semilattice, each inverse imageSa=f−1{a}is a (possibly empty) semigroup. Moreover,SbecomesgradedbyL, in the sense thatSaSb⊆Sa∧b.
Iffis onto, the semilatticeLis isomorphic to thequotientofSby the equivalence relation ~ such thatx~yif and only iff(x) =f(y). This equivalence relation is a semigroup congruence, as defined above.
Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroupS, there is a finest congruence ~ such that the quotient ofSby this equivalence relation is a semilattice. Denoting this semilattice byL, we get a homomorphismffromSontoL. As mentioned,Sbecomes graded by this semilattice.
Furthermore, the componentsSaare allArchimedean semigroups. An Archimedean semigroup is one where given any pair of elementsx,y, there exists an elementzandn> 0such thatxn=yz.
The Archimedean property follows immediately from the ordering in the semilatticeL, since with this ordering we havef(x) ≤f(y)if and only ifxn=yzfor somezandn> 0.
Thegroup of fractionsorgroup completionof a semigroupSis thegroupG=G(S)generated by the elements ofSas generators and all equationsxy=zthat hold true inSasrelations.[11]There is an obvious semigroup homomorphismj:S→G(S)that sends each element ofSto the corresponding generator. This has auniversal propertyfor morphisms fromSto a group:[12]given any groupHand any semigroup homomorphismk:S→H, there exists a uniquegroup homomorphismf:G→Hwithk=fj. We may think ofGas the "most general" group that contains a homomorphic image ofS.
An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, takeSto be the semigroup of subsets of some setXwithset-theoretic intersectionas the binary operation (this is an example of a semilattice). SinceA.A=Aholds for all elements ofS, this must be true for all generators ofG(S) as well, which is therefore thetrivial group. It is clearly necessary for embeddability thatShave thecancellation property. WhenSis commutative this condition is also sufficient[13]and theGrothendieck groupof the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups.[14][15]Anatoly Maltsevgave necessary and sufficient conditions for embeddability in 1937.[16]
Semigroup theory can be used to study some problems in the field ofpartial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as anordinary differential equationon a function space. For example, consider the following initial/boundary value problem for theheat equationon the spatialinterval(0, 1) ⊂Rand timest≥ 0:
LetX=L2((0, 1)R)be theLpspaceof square-integrable real-valued functions with domain the interval(0, 1)and letAbe the second-derivative operator withdomain
whereH2{\displaystyle H^{2}}is aSobolev space. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the spaceX:
On an heuristic level, the solution to this problem "ought" to beu(t)=exp(tA)u0.{\displaystyle u(t)=\exp(tA)u_{0}.}However, for a rigorous treatment, a meaning must be given to theexponentialoftA. As a function oft, exp(tA) is a semigroup of operators fromXto itself, taking the initial stateu0at timet= 0to the stateu(t) = exp(tA)u0at timet. The operatorAis said to be theinfinitesimal generatorof the semigroup.
The study of semigroups trailed behind that of other algebraic structures with more complex axioms such asgroupsorrings. A number of sources[17][18]attribute the first use of the term (in French) to J.-A. de Séguier inÉlements de la Théorie des Groupes Abstraits(Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton'sTheory of Groups of Finite Order.
Anton Sushkevichobtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without the rule of unique invertibility") determined the structure of finitesimple semigroupsand showed that the minimal ideal (orGreen's relationsJ-class) of a finite semigroup is simple.[18]From that point on, the foundations of semigroup theory were further laid byDavid Rees,James Alexander Green,Evgenii Sergeevich Lyapin[fr],Alfred H. CliffordandGordon Preston. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical calledSemigroup Forum(currently published bySpringer Verlag) became one of the few mathematical journals devoted entirely to semigroup theory.
Therepresentation theoryof semigroups was developed in 1963 byBoris Scheinusingbinary relationson a setAandcomposition of relationsfor the semigroup product.[19]At an algebraic conference in 1972 Schein surveyed the literature on BA, the semigroup of relations onA.[20]In 1997 Schein andRalph McKenzieproved that every semigroup is isomorphic to a transitive semigroup of binary relations.[21]
In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, likeinverse semigroups, as well as monographs focusing on applications inalgebraic automata theory, particularly for finite automata, and also infunctional analysis.
If the associativity axiom of a semigroup is dropped, the result is amagma, which is nothing more than a setMequipped with abinary operationthat is closedM×M→M.
Generalizing in a different direction, ann-ary semigroup(alson-semigroup,polyadic semigroupormultiary semigroup) is a generalization of a semigroup to a setGwith an-ary operationinstead of a binary operation.[22]The associative law is generalized as follows: ternary associativity is(abc)de=a(bcd)e=ab(cde), i.e. the stringabcdewith any three adjacent elements bracketed.n-ary associativity is a string of lengthn+ (n− 1)with anynadjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to ann-ary group.
A third generalization is thesemigroupoid, in which the requirement that the binary relation be total is lifted. As categories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities.
Infinitary generalizations of commutative semigroups have sometimes been considered by various authors.[c]
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Inmathematics, the(field) normis a particular mapping defined infield theory, which maps elements of a larger field into asubfield.
LetKbe afieldandLafiniteextension(and hence analgebraic extension) ofK.
The fieldLis then afinite-dimensionalvector spaceoverK.
Multiplication byα, an element ofL,
is aK-linear transformationof this vector space into itself.
Thenorm,NL/K(α), is defined as thedeterminantof this linear transformation.[1]
IfL/Kis aGalois extension, one may compute the norm ofα∈Las the product of all theGalois conjugatesofα:
where Gal(L/K) denotes theGalois groupofL/K.[2](Note that there may be a repetition in the terms of the product.)
For a general field extensionL/K, and nonzeroαinL, letσ1(α), ..., σn(α) be therootsof theminimal polynomialofαoverK(roots listed with multiplicity and lying in some extension field ofL); then
IfL/Kisseparable, then each root appears only once in the product (though the exponent, thedegree[L:K(α)], may still be greater than 1).
One of the basic examples of norms comes fromquadratic fieldextensionsQ(a)/Q{\displaystyle \mathbb {Q} ({\sqrt {a}})/\mathbb {Q} }wherea{\displaystyle a}is a square-free integer.
Then, the multiplication map bya{\displaystyle {\sqrt {a}}}on an elementx+y⋅a{\displaystyle x+y\cdot {\sqrt {a}}}is
The elementx+y⋅a{\displaystyle x+y\cdot {\sqrt {a}}}can be represented by the vector
since there is a direct sum decompositionQ(a)=Q⊕Q⋅a{\displaystyle \mathbb {Q} ({\sqrt {a}})=\mathbb {Q} \oplus \mathbb {Q} \cdot {\sqrt {a}}}as aQ{\displaystyle \mathbb {Q} }-vector space.
Thematrixofma{\displaystyle m_{\sqrt {a}}}is then
and the norm isNQ(a)/Q(a)=−a{\displaystyle N_{\mathbb {Q} ({\sqrt {a}})/\mathbb {Q} }({\sqrt {a}})=-a}, since it is the determinant of this matrix.
Consider thenumber fieldK=Q(2){\displaystyle K=\mathbb {Q} ({\sqrt {2}})}.
The Galois group ofK{\displaystyle K}overQ{\displaystyle \mathbb {Q} }has orderd=2{\displaystyle d=2}and is generated by the element which sends2{\displaystyle {\sqrt {2}}}to−2{\displaystyle -{\sqrt {2}}}. So the norm of1+2{\displaystyle 1+{\sqrt {2}}}is:
The field norm can also be obtained without the Galois group.
Fix aQ{\displaystyle \mathbb {Q} }-basis ofQ(2){\displaystyle \mathbb {Q} ({\sqrt {2}})}, say:
Then multiplication by the number1+2{\displaystyle 1+{\sqrt {2}}}sends
So the determinant of "multiplying by1+2{\displaystyle 1+{\sqrt {2}}}" is the determinant of the matrix which sends the vector
viz.:
The determinant of this matrix is −1.
Another easy class of examples comes from field extensions of the formQ(ap)/Q{\displaystyle \mathbb {Q} ({\sqrt[{p}]{a}})/\mathbb {Q} }where the prime factorization ofa∈Q{\displaystyle a\in \mathbb {Q} }contains nop{\displaystyle p}-th powers, forp{\displaystyle p}a fixed odd prime.
The multiplication map byap{\displaystyle {\sqrt[{p}]{a}}}of an element is
map(x)=ap⋅(a0+a1ap+a2a2p+⋯+ap−1ap−1p)=a0ap+a1a2p+a2a3p+⋯+ap−1a{\displaystyle {\begin{aligned}m_{\sqrt[{p}]{a}}(x)&={\sqrt[{p}]{a}}\cdot (a_{0}+a_{1}{\sqrt[{p}]{a}}+a_{2}{\sqrt[{p}]{a^{2}}}+\cdots +a_{p-1}{\sqrt[{p}]{a^{p-1}}})\\&=a_{0}{\sqrt[{p}]{a}}+a_{1}{\sqrt[{p}]{a^{2}}}+a_{2}{\sqrt[{p}]{a^{3}}}+\cdots +a_{p-1}a\end{aligned}}}
giving the matrix
[00⋯0a10⋯0001⋯00⋮⋮⋱⋮⋮00⋯10]{\displaystyle {\begin{bmatrix}0&0&\cdots &0&a\\1&0&\cdots &0&0\\0&1&\cdots &0&0\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\cdots &1&0\end{bmatrix}}}
The determinant gives the norm
The field norm from thecomplex numbersto thereal numberssends
to
because the Galois group ofC{\displaystyle \mathbb {C} }overR{\displaystyle \mathbb {R} }has two elements,
and taking the product yields(x+iy)(x−iy) =x2+y2.
LetL= GF(qn) be a finite extension of afinite fieldK= GF(q).
SinceL/Kis a Galois extension, ifαis inL, then the norm ofαis the product of all the Galois conjugates ofα, i.e.[3]
In this setting we have the additional properties,[4]
Several properties of the norm function hold for any finite extension.[5][6]
The normNL/K:L* →K* is agroup homomorphismfrom the multiplicative group ofLto the multiplicative group ofK, that is
Furthermore, ifainK:
Ifa∈KthenNL/K(a)=a[L:K].{\displaystyle \operatorname {N} _{L/K}(a)=a^{[L:K]}.}
Additionally, the norm behaves well intowers of fields:
ifMis a finite extension ofL, then the norm fromMtoKis just the composition of the norm fromMtoLwith the norm fromLtoK, i.e.
The norm of an element in an arbitrary field extension can be reduced to an easier computation if the degree of the field extension is already known. This is
NL/K(α)=NK(α)/K(α)[L:K(α)]{\displaystyle N_{L/K}(\alpha )=N_{K(\alpha )/K}(\alpha )^{[L:K(\alpha )]}}[6]
For example, forα=2{\displaystyle \alpha ={\sqrt {2}}}in the field extensionL=Q(2,ζ3),K=Q{\displaystyle L=\mathbb {Q} ({\sqrt {2}},\zeta _{3}),K=\mathbb {Q} }, the norm ofα{\displaystyle \alpha }is
NQ(2,ζ3)/Q(2)=NQ(2)/Q(2)[Q(2,ζ3):Q(2)]=(−2)2=4{\displaystyle {\begin{aligned}N_{\mathbb {Q} ({\sqrt {2}},\zeta _{3})/\mathbb {Q} }({\sqrt {2}})&=N_{\mathbb {Q} ({\sqrt {2}})/\mathbb {Q} }({\sqrt {2}})^{[\mathbb {Q} ({\sqrt {2}},\zeta _{3}):\mathbb {Q} ({\sqrt {2}})]}\\&=(-2)^{2}\\&=4\end{aligned}}}
since the degree of the field extensionL/K(α){\displaystyle L/K(\alpha )}is2{\displaystyle 2}.
ForOK{\displaystyle {\mathcal {O}}_{K}}thering of integersof analgebraic number fieldK{\displaystyle K}, an elementα∈OK{\displaystyle \alpha \in {\mathcal {O}}_{K}}is a unit if and only ifNK/Q(α)=±1{\displaystyle N_{K/\mathbb {Q} }(\alpha )=\pm 1}.
For instance
where
Thus, any number fieldK{\displaystyle K}whose ring of integersOK{\displaystyle {\mathcal {O}}_{K}}containsζ3{\displaystyle \zeta _{3}}has it as a unit.
The norm of analgebraic integeris again an integer, because it is equal (up to sign) to the constant term of the characteristic polynomial.
Inalgebraic number theoryone defines also norms forideals. This is done in such a way that ifIis a nonzero ideal ofOK, the ring of integers of the number fieldK,N(I) is the number of residue classes inOK/I{\displaystyle O_{K}/I}– i.e. the cardinality of thisfinite ring. Hence thisideal normis always a positive integer.
WhenIis aprincipal idealαOKthenN(I) is equal to theabsolute valueof the norm toQofα, forαanalgebraic integer.
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Inmathematics, theDedekind zeta functionof analgebraic number fieldK, generally denoted ζK(s), is a generalization of theRiemann zeta function(which is obtained in the case whereKis thefield of rational numbersQ). It can be defined as aDirichlet series, it has anEuler productexpansion, it satisfies afunctional equation, it has ananalytic continuationto ameromorphic functionon thecomplex planeCwith only asimple poleats= 1, and its values encode arithmetic data ofK. Theextended Riemann hypothesisstates that ifζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.
The Dedekind zeta function is named forRichard Dedekindwho introduced it in his supplement toPeter Gustav Lejeune Dirichlet'sVorlesungen über Zahlentheorie.[1]
LetKbe analgebraic number field. Its Dedekind zeta function is first defined forcomplex numbersswithreal partRe(s) > 1 by the Dirichlet series
whereIranges through the non-zeroidealsof thering of integersOKofKandNK/Q(I) denotes theabsolute normofI(which is equal to both theindex[OK:I] ofIinOKor equivalently thecardinalityof thequotient ringOK/I). This sum converges absolutely for all complex numbersswithreal partRe(s) > 1. In the caseK=Q, this definition reduces to that of the Riemann zeta function.
The Dedekind zeta function ofK{\displaystyle K}has an Euler product which is a product over all the non-zeroprime idealsp{\displaystyle {\mathfrak {p}}}ofOK{\displaystyle {\mathcal {O}}_{K}}
This is the expression in analytic terms of theuniqueness of prime factorization of idealsinOK{\displaystyle {\mathcal {O}}_{K}}. ForRe(s)>1,ζK(s){\displaystyle \mathrm {Re} (s)>1,\ \zeta _{K}(s)}is non-zero.
Erich Heckefirst proved thatζK(s) has an analytic continuation to a meromorphic function that is analytic at all points of the complex plane except for one simple pole ats= 1. Theresidueat that pole is given by theanalytic class number formulaand is made up of important arithmetic data involving invariants of theunit groupandclass groupofK.
The Dedekind zeta function satisfies a functional equation relating its values atsand 1 −s. Specifically, let ΔKdenote thediscriminantofK, letr1(resp.r2) denote the number of realplaces(resp. complex places) ofK, and let
and
where Γ(s) is thegamma function. Then, the functions
satisfy the functional equation
Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the fieldK. For example, theanalytic class number formularelates the residue ats= 1 to theclass numberh(K) ofK, theregulatorR(K) ofK, the numberw(K) of roots of unity inK, the absolute discriminant ofK, and the number of real and complex places ofK. Another example is ats= 0 where it has a zero whose orderris equal to therankof the unit group ofOKand the leading term is given by
It follows from the functional equation thatr=r1+r2−1{\displaystyle r=r_{1}+r_{2}-1}.
Combining the functional equation and the fact that Γ(s) is infinite at all integers less than or equal to zero yields thatζK(s) vanishes at all negative even integers. It even vanishes at all negative odd integers unlessKistotally real(i.e.r2= 0; e.g.Qor areal quadratic field). In the totally real case,Carl Ludwig Siegelshowed thatζK(s) is a non-zero rational number at negative odd integers.Stephen Lichtenbaumconjectured specific values for these rational numbers in terms of thealgebraic K-theoryofK.
For the case in whichKis anabelian extensionofQ, its Dedekind zeta function can be written as a product ofDirichlet L-functions. For example, whenKis aquadratic fieldthis shows that the ratio
is theL-functionL(s, χ), where χ is aJacobi symbolused asDirichlet character. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain DirichletL-function is an analytic formulation of thequadratic reciprocitylaw of Gauss.
In general, ifKis aGalois extensionofQwithGalois groupG, its Dedekind zeta function is theArtinL-functionof theregular representationofGand hence has a factorization in terms of ArtinL-functions ofirreducibleArtin representationsofG.
The relation with Artin L-functions shows that ifL/Kis a Galois extension thenζL(s)ζK(s){\displaystyle {\frac {\zeta _{L}(s)}{\zeta _{K}(s)}}}is holomorphic (ζK(s){\displaystyle \zeta _{K}(s)}"divides"ζL(s){\displaystyle \zeta _{L}(s)}): for general extensions the result would follow from theArtin conjecture for L-functions.[2]
Additionally,ζK(s) is theHasse–Weil zeta functionofSpecOK[3]and themotivicL-functionof themotivecoming from thecohomologyof SpecK.[4]
Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. Wieb Bosma and Bart de Smit (2002) usedGassmann triplesto give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.
Perlis (1977)showed that twonumber fieldsKandLare arithmetically equivalent if and only if all but finitely many prime numbersphave the sameinertia degreesin the two fields, i.e., ifpi{\displaystyle {\mathfrak {p}}_{i}}are the prime ideals inKlying overp, then the tuples(dimZ/pOK/pi){\displaystyle (\dim _{\mathbf {Z} /p}{\mathcal {O}}_{K}/{\mathfrak {p}}_{i})}need to be the same forKand forLfor almost allp.
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TheChebotarev density theoreminalgebraic number theorydescribes statistically the splitting ofprimesin a givenGalois extensionKof the fieldQ{\displaystyle \mathbb {Q} }ofrational numbers. Generally speaking, a prime integer will factor into severalideal primesin the ring ofalgebraic integersofK. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every primepin a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primespless than a large integerN, tends to a certain limit asNgoes to infinity. It was proved byNikolai Chebotaryovin his thesis in 1922, published in (Tschebotareff 1926).
A special case that is easier to state says that ifKis analgebraic number fieldwhich is a Galois extension ofQ{\displaystyle \mathbb {Q} }of degreen, then the prime numbers that completely split inKhave density
among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, itsFrobenius element, which is a representative of a well-definedconjugacy classin theGalois group
Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class withkelements occurs with frequency asymptotic to
WhenCarl Friedrich Gaussfirst introduced the notion ofcomplex integersZ[i], he observed that the ordinary prime numbers may factor further in this new set of integers. In fact, if a primepis congruent to 1 mod 4, then it factors into a product of two distinct prime gaussian integers, or "splits completely"; ifpis congruent to 3 mod 4, then it remains prime, or is "inert"; and ifpis 2 then it becomes a product of the square of the prime(1+i){\displaystyle (1+i)}and the invertible gaussian integer−i{\displaystyle -i}; we say that 2 "ramifies". For instance,
From this description, it appears that as one considers larger and larger primes, the frequency of a prime splitting completely approaches 1/2, and likewise for the primes that remain primes inZ[i].Dirichlet's theorem on arithmetic progressionsdemonstrates that this is indeed the case. Even though the prime numbers themselves appear rather erratically, splitting of the primes in the extension
follows a simple statistical law.
Similar statistical laws also hold for splitting of primes in thecyclotomic extensions, obtained from the field of rational numbers by adjoining a primitiveroot of unityof a given order. For example, the ordinary integer primes group into four classes, each with probability 1/4, according to their pattern of splitting in the ring of integers corresponding to the 8th roots of unity.
In this case, the field extension has degree 4 and isabelian, with the Galois group isomorphic to theKlein four-group. It turned out that the Galois group of the extension plays a key role in the pattern of splitting of primes.Georg Frobeniusestablished the framework for investigating this pattern and proved a special case of the theorem. The general statement was proved byNikolai Grigoryevich Chebotaryovin 1922.
The Chebotarev density theorem may be viewed as a generalisation ofDirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that ifN≥2is an integer andaiscoprimetoN, then the proportion of the primespcongruent toamodNis asymptotic to 1/n, wheren=φ(N) is theEuler totient function. This is a special case of the Chebotarev density theorem for theNthcyclotomic fieldK. Indeed, the Galois group ofK/Qis abelian and can be canonically identified with the group of invertibleresidue classesmodN. The splitting invariant of a primepnot dividingNis simply its residue class because the number of distinct primes into whichpsplits is φ(N)/m, where m is multiplicative order ofpmoduloN;hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime toN.
In their survey article,Lenstra & Stevenhagen (1996)give an earlier result of Frobenius in this area. SupposeKis aGalois extensionof therational number fieldQ, andP(t) a monic integer polynomial such thatKis asplitting fieldofP. It makes sense to factorisePmodulo a prime numberp. Its 'splitting type' is the list of degrees of irreducible factors ofPmodp, i.e.Pfactorizes in some fashion over theprime fieldFp. Ifnis the degree ofP, then the splitting type is apartitionΠ ofn. Considering also theGalois groupGofKoverQ, eachginGis a permutation of the roots ofPinK; in other words by choosing an ordering of α and itsalgebraic conjugates,Gisfaithfully representedas a subgroup of thesymmetric groupSn. We can writegby means of itscycle representation, which gives a 'cycle type'c(g), again a partition ofn.
Thetheorem of Frobeniusstates that for any given choice of Π the primespfor which the splitting type ofPmodpis Π has anatural densityδ, with δ equal to the proportion ofginGthat have cycle type Π.
The statement of the more generalChebotarev theoremis in terms of theFrobenius elementof a prime (ideal), which is in fact an associatedconjugacy classCof elements of theGalois groupG. If we fixCthen the theorem says that asymptotically a proportion |C|/|G| of primes have associated Frobenius element asC. WhenGis abelian the classes of course each have size 1. For the case of a non-abelian group of order 6 they have size 1, 2 and 3, and there are correspondingly (for example) 50% of primespthat have an order 2 element as their Frobenius. So these primes have residue degree 2, so they split into exactly three prime ideals in a degree 6 extension ofQwith it as Galois group.[1]
LetLbe a finite Galois extension of a number fieldKwith Galois groupG. LetXbe a subset ofGthat is stable under conjugation. The set of primesvofKthat are unramified inLand whose associated Frobenius conjugacy classFvis contained inXhas density
The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.[3]
TheGeneralized Riemann hypothesisimplies aneffective version[4]of the Chebotarev density theorem: ifL/Kis a finite Galois extension with Galois groupG, andCa union of conjugacy classes ofG, the number of unramified primes ofKof norm belowxwith Frobenius conjugacy class inCis
where the constant implied in thebig-O notationis absolute,nis the degree ofLoverQ, and Δ its discriminant.
The effective form of the Chebotarev density theory becomes much weaker without GRH. TakeLto be a finite Galois extension ofQwith Galois groupGand degreed. Takeρ{\displaystyle \rho }to be a nontrivial irreducible representation ofGof degreen, and takef(ρ){\displaystyle {\mathfrak {f}}(\rho )}to be the Artin conductor of this representation. Suppose that, forρ0{\displaystyle \rho _{0}}a subrepresentation ofρ⊗ρ{\displaystyle \rho \otimes \rho }orρ⊗ρ¯{\displaystyle \rho \otimes {\bar {\rho }}},L(ρ0,s){\displaystyle L(\rho _{0},s)}is entire; that is, the Artin conjecture is satisfied for allρ0{\displaystyle \rho _{0}}. Takeχρ{\displaystyle \chi _{\rho }}to be the character associated toρ{\displaystyle \rho }. Then there is an absolute positivec{\displaystyle c}such that, forx≥2{\displaystyle x\geq 2},
wherer{\displaystyle r}is 1 ifρ{\displaystyle \rho }is trivial and is otherwise 0, and whereβ{\displaystyle \beta }is anexceptional real zeroofL(ρ,s){\displaystyle L(\rho ,s)}; if there is no such zero, thexβ/β{\displaystyle x^{\beta }/\beta }term can be ignored. The implicit constant of this expression is absolute.[5]
The statement of the Chebotarev density theorem can be generalized to the case of an infinite Galois extensionL/Kthat is unramified outside a finite setSof primes ofK(i.e. if there is a finite setSof primes ofKsuch that any prime ofKnot inSis unramified in the extensionL/K). In this case, the Galois groupGofL/Kis aprofinite groupequipped with the Krull topology. SinceGis compact in this topology, there is a uniqueHaar measureμ onG. For every primevofKnot inSthere is an associated Frobenius conjugacy classFv. The Chebotarev density theorem in this situation can be stated as follows:[2]
This reduces to the finite case whenL/Kis finite (the Haar measure is then just the counting measure).
A consequence of this version of the theorem is that the Frobenius elements of the unramified primes ofLare dense inG.
The Chebotarev density theorem reduces the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions. Specifically, it implies that as a Galois extension ofK,Lis uniquely determined by the set of primes ofKthat split completely in it.[6]A related corollary is that if almost all prime ideals ofKsplit completely inL, then in factL=K.[7]
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The following list inmathematicscontains thefinite groupsof smallorderup togroup isomorphism.
Forn= 1, 2, … the number of nonisomorphic groups of ordernis
For labeled groups, seeOEIS:A034383.
Each group is named bySmall Groups libraryas Goi, whereois the order of the group, andiis the index used to label the group within that order.
Common group names:
The notations Znand Dihnhave the advantage thatpoint groups in three dimensionsCnand Dndo not have the same notation. There are moreisometry groupsthan these two, of the same abstract group type.
The notationG×Hdenotes thedirect productof the two groups;Gndenotes the direct product of a group with itselfntimes.G⋊Hdenotes asemidirect productwhereHactsonG; this may also depend on the choice of action ofHonG.
Abelianandsimple groupsare noted. (For groups of ordern< 60, the simple groups are precisely the cyclic groups Zn, forprimen.) The equality sign ("=") denotes isomorphism.
Theidentity elementin thecycle graphsis represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.
In the lists ofsubgroups, the trivial group and the group itself are not listed. Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses.
Angle brackets<relations> show thepresentation of a group.
The finite abelian groups are either cyclic groups, or direct products thereof; seeAbelian group. The numbers of nonisomorphic abelian groups of ordersn= 1, 2, ... are
For labeled abelian groups, seeOEIS:A034382.
The numbers of non-abelian groups, by order, are counted by (sequenceA060689in theOEIS). However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are
Small groups ofprime powerorderpnare given as follows:
Most groups of small order have a SylowpsubgroupPwith anormalp-complementNfor some primepdividing the order, so can be classified in terms of the possible primesp,p-groupsP, groupsN, and actions ofPonN. In some sense this reduces the classification of these groups to the classification ofp-groups. Some of the small groups that do not have a normalp-complement include:
The smallest order for which it isnotknown how many nonisomorphic groups there are is 2048 = 211.[7]
TheGAPcomputer algebra systemcontains apackagecalled the "Small Groups library," which provides access to descriptions of small order groups. The groups are listedup toisomorphism. At present, the library contains the following groups:[8]
It contains explicit descriptions of the available groups in computer readable format.
The smallest order for which the Small Groups library does not have information is 1024.
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Inmathematics, aCayley graph, also known as aCayley color graph,Cayley diagram,group diagram, orcolor group,[1]is agraphthat encodes the abstract structure of agroup. Its definition is suggested byCayley's theorem(named afterArthur Cayley), and uses a specifiedset of generatorsfor the group. It is a central tool incombinatorialandgeometric group theory. The structure and symmetry of Cayley graphs make them particularly good candidates for constructingexpander graphs.
LetG{\displaystyle G}be agroupandS{\displaystyle S}be agenerating setofG{\displaystyle G}. The Cayley graphΓ=Γ(G,S){\displaystyle \Gamma =\Gamma (G,S)}is anedge-coloreddirected graphconstructed as follows:[2]
Not every convention requires thatS{\displaystyle S}generate the group. IfS{\displaystyle S}is not a generating set forG{\displaystyle G}, thenΓ{\displaystyle \Gamma }isdisconnectedand each connected component represents a coset of the subgroup generated byS{\displaystyle S}.
If an elements{\displaystyle s}ofS{\displaystyle S}is its own inverse,s=s−1,{\displaystyle s=s^{-1},}then it is typically represented by an undirected edge.
The setS{\displaystyle S}is often assumed to be finite, especially ingeometric group theory, which corresponds toΓ{\displaystyle \Gamma }being locally finite andG{\displaystyle G}being finitely generated.
The setS{\displaystyle S}is sometimes assumed to besymmetric(S=S−1{\displaystyle S=S^{-1}}) and not containing the groupidentity element. In this case, the uncolored Cayley graph can be represented as a simple undirectedgraph.
The groupG{\displaystyle G}actson itself by left multiplication (seeCayley's theorem). This may be viewed as the action ofG{\displaystyle G}on its Cayley graph. Explicitly, an elementh∈G{\displaystyle h\in G}maps a vertexg∈V(Γ){\displaystyle g\in V(\Gamma )}to the vertexhg∈V(Γ).{\displaystyle hg\in V(\Gamma ).}The set of edges of the Cayley graph and their color is preserved by this action: the edge(g,gs){\displaystyle (g,gs)}is mapped to the edge(hg,hgs){\displaystyle (hg,hgs)}, both having colorcs{\displaystyle c_{s}}. In fact, allautomorphismsof the colored directed graphΓ{\displaystyle \Gamma }are of this form, so thatG{\displaystyle G}is isomorphic to thesymmetry groupofΓ{\displaystyle \Gamma }.[note 1][note 2]
The left multiplication action of a group on itself issimply transitive, in particular, Cayley graphs arevertex-transitive. The following is a kind of converse to this:
Sabidussi's Theorem—An (unlabeled and uncolored) directed graphΓ{\displaystyle \Gamma }is a Cayley graph of a groupG{\displaystyle G}if and only if it admits a simply transitive action ofG{\displaystyle G}bygraph automorphisms(i.e., preserving the set of directed edges).[5]
To recover the groupG{\displaystyle G}and the generating setS{\displaystyle S}from the unlabeled directed graphΓ{\displaystyle \Gamma }, select a vertexv1∈V(Γ){\displaystyle v_{1}\in V(\Gamma )}and label it by the identity element of the group. Then label each vertexv{\displaystyle v}ofΓ{\displaystyle \Gamma }by the unique element ofG{\displaystyle G}that mapsv1{\displaystyle v_{1}}tov.{\displaystyle v.}The setS{\displaystyle S}of generators ofG{\displaystyle G}that yieldsΓ{\displaystyle \Gamma }as the Cayley graphΓ(G,S){\displaystyle \Gamma (G,S)}is the set of labels of out-neighbors ofv1{\displaystyle v_{1}}. SinceΓ{\displaystyle \Gamma }is uncolored, it might have more directed graph automorphisms than the left multiplication maps, for example group automorphisms ofG{\displaystyle G}which permuteS{\displaystyle S}.
If one instead takes the vertices to be right cosets of a fixed subgroupH,{\displaystyle H,}one obtains a related construction, theSchreier coset graph, which is at the basis ofcoset enumerationor theTodd–Coxeter process.
Knowledge about the structure of the group can be obtained by studying theadjacency matrixof the graph and in particular applying the theorems ofspectral graph theory. Conversely, for symmetric generating sets, the spectral and representation theory ofΓ(G,S){\displaystyle \Gamma (G,S)}are directly tied together: takeρ1,…,ρk{\displaystyle \rho _{1},\dots ,\rho _{k}}a complete set of irreducible representations ofG,{\displaystyle G,}and letρi(S)=∑s∈Sρi(s){\textstyle \rho _{i}(S)=\sum _{s\in S}\rho _{i}(s)}with eigenvaluesΛi(S){\displaystyle \Lambda _{i}(S)}. Then the set of eigenvalues ofΓ(G,S){\displaystyle \Gamma (G,S)}is exactly⋃iΛi(S),{\textstyle \bigcup _{i}\Lambda _{i}(S),}where eigenvalueλ{\displaystyle \lambda }appears with multiplicitydim(ρi){\displaystyle \dim(\rho _{i})}for each occurrence ofλ{\displaystyle \lambda }as an eigenvalue ofρi(S).{\displaystyle \rho _{i}(S).}
Thegenusof a group is the minimum genus for any Cayley graph of that group.[7]
For infinite groups, thecoarse geometryof the Cayley graph is fundamental togeometric group theory. For afinitely generated group, this is independent of choice of finite set of generators, hence an intrinsic property of the group. This is only interesting for infinite groups: every finite group is coarsely equivalent to a point (or the trivial group), since one can choose as finite set of generators the entire group.
Formally, for a given choice of generators, one has theword metric(the natural distance on the Cayley graph), which determines ametric space. The coarseequivalence classof this space is an invariant of the group.
WhenS=S−1{\displaystyle S=S^{-1}}, the Cayley graphΓ(G,S){\displaystyle \Gamma (G,S)}is|S|{\displaystyle |S|}-regular, so spectral techniques may be used to analyze theexpansion propertiesof the graph. In particular for abelian groups, the eigenvalues of the Cayley graph are more easily computable and given byλχ=∑s∈Sχ(s){\textstyle \lambda _{\chi }=\sum _{s\in S}\chi (s)}with top eigenvalue equal to|S|{\displaystyle |S|}, so we may useCheeger's inequalityto bound the edge expansion ratio using the spectral gap.
Representation theory can be used to construct such expanding Cayley graphs, in the form ofKazhdan property (T). The following statement holds:[8]
For example the groupG=SL3(Z){\displaystyle G=\mathrm {SL} _{3}(\mathbb {Z} )}has property (T) and is generated byelementary matricesand this gives relatively explicit examples of expander graphs.
An integral graph is one whose eigenvalues are all integers. While the complete classification of integral graphs remains an open problem, the Cayley graphs of certain groups are always integral.
Using previous characterizations of the spectrum of Cayley graphs, note thatΓ(G,S){\displaystyle \Gamma (G,S)}is integral iff the eigenvalues ofρ(S){\displaystyle \rho (S)}are integral for every representationρ{\displaystyle \rho }ofG{\displaystyle G}.
A groupG{\displaystyle G}is Cayley integral simple (CIS) if the connected Cayley graphΓ(G,S){\displaystyle \Gamma (G,S)}is integral exactly when the symmetric generating setS{\displaystyle S}is the complement of a subgroup ofG{\displaystyle G}. A result of Ahmady, Bell, and Mohar shows that all CIS groups are isomorphic toZ/pZ,Z/p2Z{\displaystyle \mathbb {Z} /p\mathbb {Z} ,\mathbb {Z} /p^{2}\mathbb {Z} }, orZ2×Z2{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}for primesp{\displaystyle p}.[9]It is important thatS{\displaystyle S}actually generates the entire groupG{\displaystyle G}in order for the Cayley graph to be connected. (IfS{\displaystyle S}does not generateG{\displaystyle G}, the Cayley graph may still be integral, but the complement ofS{\displaystyle S}is not necessarily a subgroup.)
In the example ofG=Z/5Z{\displaystyle G=\mathbb {Z} /5\mathbb {Z} }, the symmetric generating sets (up to graph isomorphism) are
The only subgroups ofZ/5Z{\displaystyle \mathbb {Z} /5\mathbb {Z} }are the whole group and the trivial group, and the only symmetric generating setS{\displaystyle S}that produces an integral graph is the complement of the trivial group. ThereforeZ/5Z{\displaystyle \mathbb {Z} /5\mathbb {Z} }must be a CIS group.
The proof of the complete CIS classification uses the fact that every subgroup and homomorphic image of a CIS group is also a CIS group.[9]
A slightly different notion is that of a Cayley integral groupG{\displaystyle G}, in which every symmetric subsetS{\displaystyle S}produces an integral graphΓ(G,S){\displaystyle \Gamma (G,S)}. Note thatS{\displaystyle S}no longer has to generate the entire group.
The complete list of Cayley integral groups is given byZ2n×Z3m,Z2n×Z4n,Q8×Z2n,S3{\displaystyle \mathbb {Z} _{2}^{n}\times \mathbb {Z} _{3}^{m},\mathbb {Z} _{2}^{n}\times \mathbb {Z} _{4}^{n},Q_{8}\times \mathbb {Z} _{2}^{n},S_{3}}, and the dicyclic group of order12{\displaystyle 12}, wherem,n∈Z≥0{\displaystyle m,n\in \mathbb {Z} _{\geq 0}}andQ8{\displaystyle Q_{8}}is the quaternion group.[9]The proof relies on two important properties of Cayley integral groups:
Given a general groupG{\displaystyle G}, a subsetS⊆G{\displaystyle S\subseteq G}is normal ifS{\displaystyle S}is closed underconjugationby elements ofG{\displaystyle G}(generalizing the notion of a normal subgroup), andS{\displaystyle S}is Eulerian if for everys∈S{\displaystyle s\in S}, the set of elements generating the cyclic group⟨s⟩{\displaystyle \langle s\rangle }is also contained inS{\displaystyle S}.
A 2019 result by Guo, Lytkina, Mazurov, and Revin proves that the Cayley graphΓ(G,S){\displaystyle \Gamma (G,S)}is integral for any Eulerian normal subsetS⊆G{\displaystyle S\subseteq G}, using purely representation theoretic techniques.[10]
The proof of this result is relatively short: givenS{\displaystyle S}an Eulerian normal subset, selectx1,…,xt∈G{\displaystyle x_{1},\dots ,x_{t}\in G}pairwise nonconjugate so thatS{\displaystyle S}is the union of theconjugacy classesCl(xi){\displaystyle \operatorname {Cl} (x_{i})}. Then using the characterization of the spectrum of a Cayley graph, one can show the eigenvalues ofΓ(G,S){\displaystyle \Gamma (G,S)}are given by{λχ=∑i=1tχ(xi)|Cl(xi)|χ(1)}{\textstyle \left\{\lambda _{\chi }=\sum _{i=1}^{t}{\frac {\chi (x_{i})\left|\operatorname {Cl} (x_{i})\right|}{\chi (1)}}\right\}}taken over irreducible charactersχ{\displaystyle \chi }ofG{\displaystyle G}. Each eigenvalueλχ{\displaystyle \lambda _{\chi }}in this set must be an element ofQ(ζ){\displaystyle \mathbb {Q} (\zeta )}forζ{\displaystyle \zeta }a primitivemth{\displaystyle m^{th}}root of unity (wherem{\displaystyle m}must be divisible by the orders of eachxi{\displaystyle x_{i}}). Because the eigenvalues are algebraic integers, to show they are integral it suffices to show that they are rational, and it suffices to showλχ{\displaystyle \lambda _{\chi }}is fixed under any automorphismσ{\displaystyle \sigma }ofQ(ζ){\displaystyle \mathbb {Q} (\zeta )}. There must be somek{\displaystyle k}relatively prime tom{\displaystyle m}such thatσ(χ(xi))=χ(xik){\displaystyle \sigma (\chi (x_{i}))=\chi (x_{i}^{k})}for alli{\displaystyle i}, and becauseS{\displaystyle S}is both Eulerian and normal,σ(χ(xi))=χ(xj){\displaystyle \sigma (\chi (x_{i}))=\chi (x_{j})}for somej{\displaystyle j}. Sendingx↦xk{\displaystyle x\mapsto x^{k}}bijects conjugacy classes, soCl(xi){\displaystyle \operatorname {Cl} (x_{i})}andCl(xj){\displaystyle \operatorname {Cl} (x_{j})}have the same size andσ{\displaystyle \sigma }merely permutes terms in the sum forλχ{\displaystyle \lambda _{\chi }}. Thereforeλχ{\displaystyle \lambda _{\chi }}is fixed for all automorphisms ofQ(ζ){\displaystyle \mathbb {Q} (\zeta )}, soλχ{\displaystyle \lambda _{\chi }}is rational and thus integral.
Consequently, ifG=An{\displaystyle G=A_{n}}is the alternating group andS{\displaystyle S}is a set of permutations given by{(12i)±1}{\displaystyle \{(12i)^{\pm 1}\}}, then the Cayley graphΓ(An,S){\displaystyle \Gamma (A_{n},S)}is integral. (This solved a previously open problem from theKourovka Notebook.) In addition whenG=Sn{\displaystyle G=S_{n}}is the symmetric group andS{\displaystyle S}is either the set of all transpositions or the set of transpositions involving a particular element, the Cayley graphΓ(G,S){\displaystyle \Gamma (G,S)}is also integral.
Cayley graphs were first considered for finite groups byArthur Cayleyin 1878.[2]Max Dehnin his unpublished lectures on group theory from 1909–10 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of theword problemfor thefundamental groupofsurfaceswith genus ≥ 2, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point.[11]
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Inmathematics, afinitely generated moduleis amodulethat has afinitegenerating set. A finitely generated module over aringRmay also be called afiniteR-module,finite overR,[1]or amodule of finite type.
Related concepts includefinitely cogenerated modules,finitely presented modules,finitely related modulesandcoherent modulesall of which are defined below. Over aNoetherian ringthe concepts of finitely generated, finitely presented and coherent modules coincide.
A finitely generated module over afieldis simply afinite-dimensionalvector space, and a finitely generated module over theintegersis simply afinitely generated abelian group.
The leftR-moduleMis finitely generated if there exista1,a2, ...,aninMsuch that for anyxinM, there existr1,r2, ...,rninRwithx=r1a1+r2a2+ ... +rnan.
Theset{a1,a2, ...,an} is referred to as agenerating setofMin this case. A finite generating set need not be a basis, since it need not be linearly independent overR. What is true is:Mis finitely generated if and only if there is a surjectiveR-linear map:
for somen; in other words,Mis aquotientof afree moduleof finite rank.
If a setSgenerates a module that is finitely generated, then there is a finite generating set that is included inS, since only finitely many elements inSare needed to express the generators in any finite generating set, and these finitely many elements form a generating set. However, it may occur thatSdoes not contain any finite generating set of minimalcardinality. For example the set of theprime numbersis a generating set ofZ{\displaystyle \mathbb {Z} }viewed asZ{\displaystyle \mathbb {Z} }-module, and a generating set formed from prime numbers has at least two elements, while thesingleton{1}is also a generating set.
In the case where themoduleMis avector spaceover afieldR, and the generating set islinearly independent,niswell-definedand is referred to as thedimensionofM(well-definedmeans that anylinearly independentgenerating set hasnelements: this is thedimension theorem for vector spaces).
Any module is the union of thedirected setof its finitely generated submodules.
A moduleMis finitely generated if and only if any increasing chainMiof submodules with unionMstabilizes: i.e., there is someisuch thatMi=M. This fact withZorn's lemmaimplies that every nonzero finitely generated module admitsmaximal submodules. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the moduleMis called aNoetherian module.
Everyhomomorphic imageof a finitely generated module is finitely generated. In general,submodulesof finitely generated modules need not be finitely generated. As an example, consider the ringR=Z[X1,X2, ...] of allpolynomialsincountably manyvariables.Ritself is a finitely generatedR-module (with {1} as generating set). Consider the submoduleKconsisting of all those polynomials with zero constant term. Since every polynomial contains only finitely many terms whose coefficients are non-zero, theR-moduleKis not finitely generated.
In general, a module is said to beNoetherianif every submodule is finitely generated. A finitely generated module over aNoetherian ringis a Noetherian module (and indeed this property characterizes Noetherian rings): A module over a Noetherian ring is finitely generated if and only if it is a Noetherian module. This resembles, but is not exactlyHilbert's basis theorem, which states that the polynomial ringR[X] over a Noetherian ringRis Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring.
More generally, an algebra (e.g., ring) that is a finitely generated module is afinitely generated algebra. Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (Seeintegral elementfor more.)
Let 0 →M′ →M→M′′ → 0 be anexact sequenceof modules. ThenMis finitely generated ifM′,M′′ are finitely generated. There are some partial converses to this. IfMis finitely generated andM′′ is finitely presented (which is stronger than finitely generated; see below), thenM′ is finitely generated. Also,Mis Noetherian (resp. Artinian) if and only ifM′,M′′ are Noetherian (resp. Artinian).
LetBbe a ring andAits subring such thatBis afaithfully flatrightA-module. Then a leftA-moduleFis finitely generated (resp. finitely presented) if and only if theB-moduleB⊗AFis finitely generated (resp. finitely presented).[2]
For finitely generated modules over a commutative ringR,Nakayama's lemmais fundamental. Sometimes, the lemma allows one to prove finite dimensional vector spaces phenomena for finitely generated modules. For example, iff:M→Mis asurjectiveR-endomorphism of a finitely generated moduleM, thenfis alsoinjective, and hence is anautomorphismofM.[3]This says simply thatMis aHopfian module. Similarly, anArtinian moduleMiscoHopfian: any injective endomorphismfis also a surjective endomorphism.[4]TheForster–Swan theoremgives an upper bound for the minimal number of generators of a finitely generated moduleMover a commutative Noetherian ring.
AnyR-module is aninductive limitof finitely generatedR-submodules. This is useful for weakening an assumption to the finite case (e.g., thecharacterization of flatnesswith theTor functor).
An example of a link between finite generation andintegral elementscan be found in commutative algebras. To say that a commutative algebraAis afinitely generated ringoverRmeans that there exists a set of elementsG= {x1, ...,xn}ofAsuch that the smallest subring ofAcontainingGandRisAitself. Because the ring product may be used to combine elements, more than justR-linear combinations of elements ofGare generated. For example, apolynomial ringR[x] is finitely generated by {1,x} as a ring,but not as a module. IfAis a commutative algebra (with unity) overR, then the following two statements are equivalent:[5]
LetMbe a finitely generated module over an integral domainAwith the field of fractionsK. Then the dimensiondimK(M⊗AK){\displaystyle \operatorname {dim} _{K}(M\otimes _{A}K)}is called thegeneric rankofMoverA. This number is the same as the number of maximalA-linearly independent vectors inMor equivalently the rank of a maximal free submodule ofM(cf.Rank of an abelian group). Since(M/F)(0)=M(0)/F(0)=0{\displaystyle (M/F)_{(0)}=M_{(0)}/F_{(0)}=0},M/F{\displaystyle M/F}is atorsion module. WhenAis Noetherian, bygeneric freeness, there is an elementf(depending onM) such thatM[f−1]{\displaystyle M[f^{-1}]}is a freeA[f−1]{\displaystyle A[f^{-1}]}-module. Then the rank of this free module is the generic rank ofM.
Now suppose the integral domainAis anN{\displaystyle \mathbb {N} }-graded algebraover a fieldkgenerated by finitely many homogeneous elements of degreesdi{\displaystyle d_{i}}. SupposeMis graded as well and letPM(t)=∑(dimkMn)tn{\displaystyle P_{M}(t)=\sum (\operatorname {dim} _{k}M_{n})t^{n}}be thePoincaré seriesofM.
By theHilbert–Serre theorem, there is a polynomialFsuch thatPM(t)=F(t)∏(1−tdi)−1{\displaystyle P_{M}(t)=F(t)\prod (1-t^{d_{i}})^{-1}}. ThenF(1){\displaystyle F(1)}is the generic rank ofM.[6]
A finitely generated module over aprincipal ideal domainistorsion-freeif and only if it is free. This is a consequence of thestructure theorem for finitely generated modules over a principal ideal domain, the basic form of which says a finitely generated module over a PID is a direct sum of a torsion module and a free module. But it can also be shown directly as follows: letMbe a torsion-free finitely generated module over a PIDAandFa maximal free submodule. Letfbe inAsuch thatfM⊂F{\displaystyle fM\subset F}. ThenfM{\displaystyle fM}is free since it is a submodule of a free module andAis a PID. But nowf:M→fM{\displaystyle f:M\to fM}is an isomorphism sinceMis torsion-free.
By the same argument as above, a finitely generated module over aDedekind domainA(or more generally asemi-hereditary ring) is torsion-free if and only if it isprojective; consequently, a finitely generated module overAis a direct sum of a torsion module and a projective module. A finitely generated projective module over a Noetherian integral domain has constant rank and so the generic rank of a finitely generated module overAis the rank of its projective part.
The following conditions are equivalent toMbeing finitely generated (f.g.):
From these conditions it is easy to see that being finitely generated is a property preserved byMorita equivalence. The conditions are also convenient to define adualnotion of afinitely cogenerated moduleM. The following conditions are equivalent to a module being finitely cogenerated (f.cog.):
Both f.g. modules and f.cog. modules have interesting relationships to Noetherian and Artinian modules, and theJacobson radicalJ(M) andsoclesoc(M) of a module. The following facts illustrate the duality between the two conditions. For a moduleM:
Finitely cogenerated modules must have finiteuniform dimension. This is easily seen by applying the characterization using the finitely generated essential socle. Somewhat asymmetrically, finitely generated modulesdo notnecessarily have finite uniform dimension. For example, an infinite direct product of nonzero rings is a finitely generated (cyclic!) module over itself, however it clearly contains an infinite direct sum of nonzero submodules. Finitely generated modulesdo notnecessarily have finiteco-uniform dimensioneither: any ringRwith unity such thatR/J(R) is not a semisimple ring is a counterexample.
Another formulation is this: a finitely generated moduleMis one for which there is anepimorphismmappingRkontoM:
Suppose now there is an epimorphism,
for a moduleMand free moduleF.
Over any ringR, coherent modules are finitely presented, and finitely presented modules are both finitely generated and finitely related. For aNoetherian ringR, finitely generated, finitely presented, and coherent are equivalent conditions on a module.
Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and a finitely related flat module is projective.
It is true also that the following conditions are equivalent for a ringR:
Although coherence seems like a more cumbersome condition than finitely generated or finitely presented, it is nicer than them since thecategoryof coherent modules is anabelian category, while, in general, neither finitely generated nor finitely presented modules form an abelian category.
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Innumber theory, given aprime numberp,[note 1]thep-adic numbersform an extension of therational numberswhich is distinct from thereal numbers, though with some similar properties;p-adic numbers can be written in a form similar to (possiblyinfinite)decimals, but with digits based on a prime numberprather than ten, and extending to the left rather than to the right.
For example, comparing the expansion of the rational number15{\displaystyle {\tfrac {1}{5}}}inbase3vs. the3-adic expansion,
Formally, given a prime numberp, ap-adic number can be defined as aseries
wherekis aninteger(possibly negative), and eachai{\displaystyle a_{i}}is an integer such that0≤ai<p.{\displaystyle 0\leq a_{i}<p.}Ap-adic integeris ap-adic number such thatk≥0.{\displaystyle k\geq 0.}
In general the series that represents ap-adic number is notconvergentin the usual sense, but it is convergent for thep-adic absolute value|s|p=p−k,{\displaystyle |s|_{p}=p^{-k},}wherekis the least integerisuch thatai≠0{\displaystyle a_{i}\neq 0}(if allai{\displaystyle a_{i}}are zero, one has the zerop-adic number, which has0as itsp-adic absolute value).
Every rational number can be uniquely expressed as the sum of a series as above, with respect to thep-adic absolute value. This allows considering rational numbers as specialp-adic numbers, and alternatively defining thep-adic numbers as thecompletionof the rational numbers for thep-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
p-adic numbers were first described byKurt Henselin 1897,[1]though, with hindsight, some ofErnst Kummer'searlier work can be interpreted as implicitly usingp-adic numbers.[note 2]
Roughly speaking,modular arithmeticmodulo a positive integernconsists of "approximating" every integer by the remainder of itsdivisionbyn, called itsresidue modulon. The main property of modular arithmetic is that the residue modulonof the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulon. If one knows that the absolute value of the result is less thann/2, this allows a computation of the result which does not involve any integer larger thann.
For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying theChinese remainder theoremfor recovering the result modulo the product of the moduli.
Another method discovered byKurt Henselconsists of using a prime modulusp, and applyingHensel's lemmafor recovering iteratively the result modulop2,p3,…,pn,…{\displaystyle p^{2},p^{3},\ldots ,p^{n},\ldots }If the process is continued infinitely, this provides eventually a result which is ap-adic number.
The theory ofp-adic numbers is fundamentally based on the two following lemmas:
Every nonzero rational number can be writtenpvmn,{\textstyle p^{v}{\frac {m}{n}},}wherev,m, andnare integers and neithermnornis divisible byp.The exponentvis uniquely determined by the rational number and is called itsp-adic valuation(this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from thefundamental theorem of arithmetic.
Every nonzero rational numberrof valuationvcan be uniquely writtenr=apv+s,{\displaystyle r=ap^{v}+s,}wheresis a rational number of valuation greater thanv, andais an integer such that0<a<p.{\displaystyle 0<a<p.}
The proof of this lemma results frommodular arithmetic: By the above lemma,r=pvmn,{\textstyle r=p^{v}{\frac {m}{n}},}wheremandnare integerscoprimewithp.
ByBézout's lemma, there exist integersaandb, with0≤a<p{\displaystyle 0\leq a<p}, such thatm=an+bp.{\displaystyle m=an+bp.}Settings=b/n{\displaystyle s=b/n}(henceval(s)≥0{\displaystyle {\rm {val}}(s)\geq 0}), we have
To show the uniqueness of this representation, observe that ifr=a′pv+pv+1s′,{\displaystyle r=a'p^{v}+p^{v+1}s',}with0≤a′<p{\displaystyle 0\leq a'<p}andval(s′)≥0{\displaystyle {\rm {val}}(s')\geq 0},
there holds by difference(a−a′)+p(s−s′)=0,{\displaystyle (a-a')+p(s-s')=0,}with|a−a′|<p{\displaystyle |a-a'|<p}andval(s−s′)≥0{\displaystyle {\rm {val}}(s-s')\geq 0}.
Writes−s′=c/d{\displaystyle s-s'=c/d}, wheredis coprime top; then(a−a′)d+pc=0{\displaystyle (a-a')d+pc=0}, which is possible only ifa−a′=0{\displaystyle a-a'=0}andc=0{\displaystyle c=0}.
Hencea=a′{\displaystyle a=a'}ands=s′{\displaystyle s=s'}.
The above process can be iterated starting fromsinstead ofr, giving the following.
Given a nonzero rational numberrof valuationvand a positive integerk, there are a rational numbersk{\displaystyle s_{k}}of nonnegative valuation andkuniquely defined nonnegative integersa0,…,ak−1{\displaystyle a_{0},\ldots ,a_{k-1}}less thanpsuch thata0>0{\displaystyle a_{0}>0}and
Thep-adic numbers are essentially obtained by continuing this infinitely to produce aninfinite series.
Thep-adic numbers are commonly defined by means ofp-adic series.
Ap-adic seriesis aformal power seriesof the form
wherev{\displaystyle v}is an integer and theri{\displaystyle r_{i}}are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator ofri{\displaystyle r_{i}}is not divisible byp).
Every rational number may be viewed as ap-adic series with a single nonzero term, consisting of its factorization of the formpknd,{\displaystyle p^{k}{\tfrac {n}{d}},}withnanddboth coprime withp.
Twop-adic series∑i=v∞ripi{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i}}and∑i=w∞sipi{\textstyle \sum _{i=w}^{\infty }s_{i}p^{i}}areequivalentif there is an integerNsuch that, for every integern>N,{\displaystyle n>N,}the rational number
is zero or has ap-adic valuation greater thann.
Ap-adic series∑i=v∞aipi{\textstyle \sum _{i=v}^{\infty }a_{i}p^{i}}isnormalizedif either allai{\displaystyle a_{i}}are integers such that0≤ai<p,{\displaystyle 0\leq a_{i}<p,}andav>0,{\displaystyle a_{v}>0,}or allai{\displaystyle a_{i}}are zero. In the latter case, the series is called thezero series.
Everyp-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see§ Normalization of ap-adic series, below.
In other words, the equivalence ofp-adic series is anequivalence relation, and eachequivalence classcontains exactly one normalizedp-adic series.
The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence ofp-adic series. That is, denoting the equivalence with~, ifS,TandUare nonzerop-adic series such thatS∼T,{\displaystyle S\sim T,}one has
Thep-adic numbers are often defined as the equivalence classes ofp-adic series, in a similar way as the definition of the real numbers as equivalence classes ofCauchy sequences. The uniqueness property of normalization, allows uniquely representing anyp-adic number by the corresponding normalizedp-adic series. The compatibility of the series equivalence leads almost immediately to basic properties ofp-adic numbers:
Starting with the series∑i=v∞ripi,{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i},}the first above lemma allows getting an equivalent series such that thep-adic valuation ofrv{\displaystyle r_{v}}is zero. For that, one considers the first nonzerori.{\displaystyle r_{i}.}If itsp-adic valuation is zero, it suffices to changevintoi, that is to start the summation fromv. Otherwise, thep-adic valuation ofri{\displaystyle r_{i}}isj>0,{\displaystyle j>0,}andri=pjsi{\displaystyle r_{i}=p^{j}s_{i}}where the valuation ofsi{\displaystyle s_{i}}is zero; so, one gets an equivalent series by changingri{\displaystyle r_{i}}to0andri+j{\displaystyle r_{i+j}}tori+j+si.{\displaystyle r_{i+j}+s_{i}.}Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation ofrv{\displaystyle r_{v}}is zero.
Then, if the series is not normalized, consider the first nonzerori{\displaystyle r_{i}}that is not an integer in the interval[0,p−1].{\displaystyle [0,p-1].}The second above lemma allows writing itri=ai+psi;{\displaystyle r_{i}=a_{i}+ps_{i};}one gets n equivalent series by replacingri{\displaystyle r_{i}}withai,{\displaystyle a_{i},}and addingsi{\displaystyle s_{i}}tori+1.{\displaystyle r_{i+1}.}Iterating this process, possibly infinitely many times, provides eventually the desired normalizedp-adic series.
There are several equivalent definitions ofp-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions usecompletionof adiscrete valuation ring(see§ p-adic integers),completion of a metric space(see§ Topological properties), orinverse limits(see§ Modular properties).
Ap-adic number can be defined as anormalizedp-adic series. Since there are other equivalent definitions that are commonly used, one says often that a normalizedp-adic seriesrepresentsap-adic number, instead of saying that itisap-adic number.
One can say also that anyp-adic series represents ap-adic number, since everyp-adic series is equivalent to a unique normalizedp-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) ofp-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations onp-adic numbers, since the series operations are compatible with equivalence ofp-adic series.
With these operations,p-adic numbers form afieldcalled thefield ofp-adic numbersand denotedQp{\displaystyle \mathbb {Q} _{p}}orQp.{\displaystyle \mathbf {Q} _{p}.}There is a uniquefield homomorphismfrom the rational numbers into thep-adic numbers, which maps a rational number to itsp-adic expansion. Theimageof this homomorphism is commonly identified with the field of rational numbers. This allows considering thep-adic numbers as anextension fieldof the rational numbers, and the rational numbers as asubfieldof thep-adic numbers.
Thevaluationof a nonzerop-adic numberx, commonly denotedvp(x),{\displaystyle v_{p}(x),}is the exponent ofpin the first nonzero term of everyp-adic series that representsx. By convention,vp(0)=∞;{\displaystyle v_{p}(0)=\infty ;}that is, the valuation of zero is∞.{\displaystyle \infty .}This valuation is adiscrete valuation. The restriction of this valuation to the rational numbers is thep-adic valuation ofQ,{\displaystyle \mathbb {Q} ,}that is, the exponentvin the factorization of a rational number asndpv,{\displaystyle {\tfrac {n}{d}}p^{v},}with bothnanddcoprimewithp.
Thep-adic integersare thep-adic numbers with a nonnegative valuation.
Ap{\displaystyle p}-adic integer can be represented as a sequence
of residuesxe{\displaystyle x_{e}}modpe{\displaystyle p^{e}}for each integere{\displaystyle e}, satisfying the compatibility relationsxi≡xj(modpi){\displaystyle x_{i}\equiv x_{j}~(\operatorname {mod} p^{i})}fori<j{\displaystyle i<j}.
Everyintegeris ap{\displaystyle p}-adic integer (including zero, since0<∞{\displaystyle 0<\infty }). The rational numbers of the formndpk{\textstyle {\tfrac {n}{d}}p^{k}}withd{\displaystyle d}coprime withp{\displaystyle p}andk≥0{\displaystyle k\geq 0}are alsop{\displaystyle p}-adic integers (for the reason thatd{\displaystyle d}has an inverse modpe{\displaystyle p^{e}}for everye{\displaystyle e}).
Thep-adic integers form acommutative ring, denotedZp{\displaystyle \mathbb {Z} _{p}}orZp{\displaystyle \mathbf {Z} _{p}}, that has the following properties.
The last property provides a definition of thep-adic numbers that is equivalent to the above one: the field of thep-adic numbers is thefield of fractionsof the completion of the localization of the integers at the prime ideal generated byp.
Thep-adic valuation allows defining anabsolute valueonp-adic numbers: thep-adic absolute value of a nonzerop-adic numberxis
wherevp(x){\displaystyle v_{p}(x)}is thep-adic valuation ofx. Thep-adic absolute value of0{\displaystyle 0}is|0|p=0.{\displaystyle |0|_{p}=0.}This is an absolute value that satisfies thestrong triangle inequalitysince, for everyxandyone has
Moreover, if|x|p≠|y|p,{\displaystyle |x|_{p}\neq |y|_{p},}one has|x+y|p=max(|x|p,|y|p).{\displaystyle |x+y|_{p}=\max(|x|_{p},|y|_{p}).}
This makes thep-adic numbers ametric space, and even anultrametric space, with thep-adic distance defined bydp(x,y)=|x−y|p.{\displaystyle d_{p}(x,y)=|x-y|_{p}.}
As a metric space, thep-adic numbers form thecompletionof the rational numbers equipped with thep-adic absolute value. This provides another way for defining thep-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from everyCauchy sequencea subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of thepartial sumsof ap-adic series, and thus a unique normalizedp-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalizedp-adic series instead of equivalence classes of Cauchy sequences).
As the metric is defined from a discrete valuation, everyopen ballis alsoclosed. More precisely, the open ballBr(x)={y∣dp(x,y)<r}{\displaystyle B_{r}(x)=\{y\mid d_{p}(x,y)<r\}}equals the closed ballBp−v[x]={y∣dp(x,y)≤p−v},{\displaystyle B_{p^{-v}}[x]=\{y\mid d_{p}(x,y)\leq p^{-v}\},}wherevis the least integer such thatp−v<r.{\displaystyle p^{-v}<r.}Similarly,Br[x]=Bp−w(x),{\displaystyle B_{r}[x]=B_{p^{-w}}(x),}wherewis the greatest integer such thatp−w>r.{\displaystyle p^{-w}>r.}
This implies that thep-adic numbers form alocally compact space(locally compact field), and thep-adic integers—that is, the ballB1[0]=Bp(0){\displaystyle B_{1}[0]=B_{p}(0)}—form acompact space.
Thedecimal expansionof a positiverational numberr{\displaystyle r}is its representation as aseries
wherek{\displaystyle k}is an integer and eachai{\displaystyle a_{i}}is also anintegersuch that0≤ai<10.{\displaystyle 0\leq a_{i}<10.}This expansion can be computed bylong divisionof the numerator by the denominator, which is itself based on the following theorem: Ifr=nd{\displaystyle r={\tfrac {n}{d}}}is a rational number such that10k≤r<10k+1,{\displaystyle 10^{k}\leq r<10^{k+1},}there is an integera{\displaystyle a}such that0<a<10,{\displaystyle 0<a<10,}andr=a10k+r′,{\displaystyle r=a\,10^{k}+r',}withr′<10k.{\displaystyle r'<10^{k}.}The decimal expansion is obtained by repeatedly applying this result to the remainderr′{\displaystyle r'}which in the iteration assumes the role of the original rational numberr{\displaystyle r}.
Thep-adic expansionof a rational number is defined similarly, but with a different division step. More precisely, given a fixedprime numberp{\displaystyle p}, every nonzero rational numberr{\displaystyle r}can be uniquely written asr=pknd,{\displaystyle r=p^{k}{\tfrac {n}{d}},}wherek{\displaystyle k}is a (possibly negative) integer,n{\displaystyle n}andd{\displaystyle d}arecoprime integersboth coprime withp{\displaystyle p}, andd{\displaystyle d}is positive. The integerk{\displaystyle k}is thep-adic valuationofr{\displaystyle r}, denotedvp(r),{\displaystyle v_{p}(r),}andp−k{\displaystyle p^{-k}}is itsp-adic absolute value, denoted|r|p{\displaystyle |r|_{p}}(the absolute value is small when the valuation is large). The division step consists of writing
wherea{\displaystyle a}is an integer such that0≤a<p,{\displaystyle 0\leq a<p,}andr′{\displaystyle r'}is either zero, or a rational number such that|r′|p<p−k{\displaystyle |r'|_{p}<p^{-k}}(that is,vp(r′)>k{\displaystyle v_{p}(r')>k}).
Thep{\displaystyle p}-adic expansionofr{\displaystyle r}is theformal power series
obtained by repeating indefinitely theabovedivision step on successive remainders. In ap-adic expansion, allai{\displaystyle a_{i}}are integers such that0≤ai<p.{\displaystyle 0\leq a_{i}<p.}
Ifr=pkn1{\displaystyle r=p^{k}{\tfrac {n}{1}}}withn>0{\displaystyle n>0}, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation ofr{\displaystyle r}inbase-p.
The existence and the computation of thep-adic expansion of a rational number results fromBézout's identityin the following way. If, as above,r=pknd,{\displaystyle r=p^{k}{\tfrac {n}{d}},}andd{\displaystyle d}andp{\displaystyle p}are coprime, there exist integerst{\displaystyle t}andu{\displaystyle u}such thattd+up=1.{\displaystyle td+up=1.}So
Then, theEuclidean divisionofnt{\displaystyle nt}byp{\displaystyle p}gives
with0≤a<p.{\displaystyle 0\leq a<p.}This gives the division step as
so that in the iteration
is the new rational number.
The uniqueness of the division step and of the wholep-adic expansion is easy: ifpka1+pk+1s1=pka2+pk+1s2,{\displaystyle p^{k}a_{1}+p^{k+1}s_{1}=p^{k}a_{2}+p^{k+1}s_{2},}one hasa1−a2=p(s2−s1).{\displaystyle a_{1}-a_{2}=p(s_{2}-s_{1}).}This meansp{\displaystyle p}dividesa1−a2.{\displaystyle a_{1}-a_{2}.}Since0≤a1<p{\displaystyle 0\leq a_{1}<p}and0≤a2<p,{\displaystyle 0\leq a_{2}<p,}the following must be true:0≤a1{\displaystyle 0\leq a_{1}}anda2<p.{\displaystyle a_{2}<p.}Thus, one gets−p<a1−a2<p,{\displaystyle -p<a_{1}-a_{2}<p,}and sincep{\displaystyle p}dividesa1−a2{\displaystyle a_{1}-a_{2}}it must be thata1=a2.{\displaystyle a_{1}=a_{2}.}
Thep-adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of aconvergent serieswith thep-adic absolute value.
In the standardp-adic notation, the digits are written in the same order as in astandard base-psystem, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side.
Thep-adic expansion of a rational number is eventuallyperiodic.Conversely, a series∑i=k∞aipi,{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i},}with0≤ai<p{\displaystyle 0\leq a_{i}<p}converges (for thep-adic absolute value) to a rational numberif and only ifit is eventually periodic; in this case, the series is thep-adic expansion of that rational number. Theproofis similar to that of the similar result forrepeating decimals.
Let us compute the 5-adic expansion of13.{\displaystyle {\tfrac {1}{3}}.}Bézout's identity for 5 and the denominator 3 is2⋅3+(−1)⋅5=1{\displaystyle 2\cdot 3+(-1)\cdot 5=1}(for larger examples, this can be computed with theextended Euclidean algorithm). Thus
For the next step, one has to expand−1/3{\displaystyle -1/3}(the factor 5 has to be viewed as a "shift" of thep-adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded). To expand−1/3{\displaystyle -1/3}, we start from the same Bézout's identity and multiply it by−1{\displaystyle -1}, giving
The "integer part"−2{\displaystyle -2}is not in the right interval. So, one has to useEuclidean divisionby5{\displaystyle 5}for getting−2=3−1⋅5,{\displaystyle -2=3-1\cdot 5,}giving
and the expansion in the first step becomes
Similarly, one has
and
As the "remainder"−13{\displaystyle -{\tfrac {1}{3}}}has already been found, the process can be continued easily, giving coefficients3{\displaystyle 3}foroddpowers of five, and1{\displaystyle 1}forevenpowers.
Or in the standard 5-adic notation
with theellipsis…{\displaystyle \ldots }on the left hand side.
It is possible to use apositional notationsimilar to that which is used to represent numbers inbasep.
Let∑i=k∞aipi{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i}}be a normalizedp-adic series, i.e. eachai{\displaystyle a_{i}}is an integer in the interval[0,p−1].{\displaystyle [0,p-1].}One can suppose thatk≤0{\displaystyle k\leq 0}by settingai=0{\displaystyle a_{i}=0}for0≤i<k{\displaystyle 0\leq i<k}(ifk>0{\displaystyle k>0}), and adding the resulting zero terms to the series.
Ifk≥0,{\displaystyle k\geq 0,}the positional notation consists of writing theai{\displaystyle a_{i}}consecutively, ordered by decreasing values ofi, often withpappearing on the right as an index:
So, the computation of theexample aboveshows that
and
Whenk<0,{\displaystyle k<0,}a separating dot is added before the digits with negative index, and, if the indexpis present, it appears just after the separating dot. For example,
and
If ap-adic representation is finite on the left (that is,ai=0{\displaystyle a_{i}=0}for large values ofi), then it has the value of a nonnegative rational number of the formnpv,{\displaystyle np^{v},}withn,v{\displaystyle n,v}integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation inbasep. For these rational numbers, the two representations are the same.
Thequotient ringZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}may be identified with theringZ/pnZ{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }of the integersmodulopn.{\displaystyle p^{n}.}This can be shown by remarking that everyp-adic integer, represented by its normalizedp-adic series, is congruent modulopn{\displaystyle p^{n}}with itspartial sum∑i=0n−1aipi,{\textstyle \sum _{i=0}^{n-1}a_{i}p^{i},}whose value is an integer in the interval[0,pn−1].{\displaystyle [0,p^{n}-1].}A straightforward verification shows that this defines aring isomorphismfromZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}toZ/pnZ.{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} .}
Theinverse limitof the ringsZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}is defined as the ring formed by the sequencesa0,a1,…{\displaystyle a_{0},a_{1},\ldots }such thatai∈Z/piZ{\displaystyle a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z} }andai≡ai+1(modpi){\textstyle a_{i}\equiv a_{i+1}{\pmod {p^{i}}}}for everyi.
The mapping that maps a normalizedp-adic series to the sequence of its partial sums is a ring isomorphism fromZp{\displaystyle \mathbb {Z} _{p}}to the inverse limit of theZp/pnZp.{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}.}This provides another way for definingp-adic integers (up toan isomorphism).
This definition ofp-adic integers is specially useful for practical computations, as allowing buildingp-adic integers by successive approximations.
For example, for computing thep-adic (multiplicative) inverse of an integer, one can useNewton's method, starting from the inverse modulop; then, each Newton step computes the inverse modulopn2{\textstyle p^{n^{2}}}from the inverse modulopn.{\textstyle p^{n}.}
The same method can be used for computing thep-adicsquare rootof an integer that is aquadratic residuemodulop. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found inZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}. Applying Newton's method to find the square root requirespn{\textstyle p^{n}}to be larger than twice the given integer, which is quickly satisfied.
Hensel liftingis a similar method that allows to "lift" the factorization modulopof a polynomial with integer coefficients to a factorization modulopn{\textstyle p^{n}}for large values ofn. This is commonly used bypolynomial factorizationalgorithms.
There are several different conventions for writingp-adic expansions. So far this article has used a notation forp-adic expansions in whichpowersofpincrease from right to left. With this right-to-left notation the 3-adic expansion of15,{\displaystyle {\tfrac {1}{5}},}for example, is written as
When performing arithmetic in this notation, digits arecarriedto the left. It is also possible to writep-adic expansions so that the powers ofpincrease from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of15{\displaystyle {\tfrac {1}{5}}}is
p-adic expansions may be written withother sets of digitsinstead of{0, 1, ...,p− 1}. For example, the3-adic expansion of15{\displaystyle {\tfrac {1}{5}}}can be written usingbalanced ternarydigits{1, 0, 1}, with1representing negative one, as
In fact any set ofpintegers which are in distinctresidue classesmodulopmay be used asp-adic digits. In number theory,Teichmüller representativesare sometimes used as digits.[2]
Quote notationis a variant of thep-adic representation ofrational numbersthat was proposed in 1979 byEric HehnerandNigel Horspoolfor implementing on computers the (exact) arithmetic with these numbers.[3]
BothZp{\displaystyle \mathbb {Z} _{p}}andQp{\displaystyle \mathbb {Q} _{p}}areuncountableand have thecardinality of the continuum.[4]ForZp,{\displaystyle \mathbb {Z} _{p},}this results from thep-adic representation, which defines abijectionofZp{\displaystyle \mathbb {Z} _{p}}on thepower set{0,…,p−1}N.{\displaystyle \{0,\ldots ,p-1\}^{\mathbb {N} }.}ForQp{\displaystyle \mathbb {Q} _{p}}this results from its expression as acountably infiniteunionof copies ofZp{\displaystyle \mathbb {Z} _{p}}:
Qp{\displaystyle \mathbb {Q} _{p}}containsQ{\displaystyle \mathbb {Q} }and is a field ofcharacteristic0.
Because0can be written as sum of squares,[5]Qp{\displaystyle \mathbb {Q} _{p}}cannot be turned into anordered field.
The field ofreal numbersR{\displaystyle \mathbb {R} }has only a single properalgebraic extension: thecomplex numbersC{\displaystyle \mathbb {C} }. In other words, thisquadratic extensionis alreadyalgebraically closed. By contrast, thealgebraic closureofQp{\displaystyle \mathbb {Q} _{p}}, denotedQp¯,{\displaystyle {\overline {\mathbb {Q} _{p}}},}has infinite degree,[6]that is,Qp{\displaystyle \mathbb {Q} _{p}}has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of thep-adic valuation toQp¯,{\displaystyle {\overline {\mathbb {Q} _{p}}},}the latter is not (metrically) complete.[7][8]Its (metric) completion is calledCp{\displaystyle \mathbb {C} _{p}}orΩp{\displaystyle \Omega _{p}}.[8][9]Here an end is reached, asCp{\displaystyle \mathbb {C} _{p}}is algebraically closed.[8][10]However unlikeC{\displaystyle \mathbb {C} }this field is notlocally compact.[9]
Cp{\displaystyle \mathbb {C} _{p}}andC{\displaystyle \mathbb {C} }are isomorphic as rings,[11]so we may regardCp{\displaystyle \mathbb {C} _{p}}asC{\displaystyle \mathbb {C} }endowed with an exotic metric. The proof of existence of such a field isomorphism relies on theaxiom of choice, and does not provide an explicit example of such an isomorphism (that is, it is notconstructive).
IfK{\displaystyle K}is any finiteGalois extensionofQp,{\displaystyle \mathbb {Q} _{p},}theGalois groupGal(K/Qp){\displaystyle \operatorname {Gal} \left(K/\mathbb {Q} _{p}\right)}issolvable. Thus, the Galois groupGal(Qp¯/Qp){\displaystyle \operatorname {Gal} \left({\overline {\mathbb {Q} _{p}}}/\mathbb {Q} _{p}\right)}isprosolvable.
Qp{\displaystyle \mathbb {Q} _{p}}contains then-thcyclotomic field(n> 2) if and only ifn|p− 1.[12]For instance, then-th cyclotomic field is a subfield ofQ13{\displaystyle \mathbb {Q} _{13}}if and only ifn= 1, 2, 3, 4, 6, or12. In particular, there is no multiplicativep-torsioninQp{\displaystyle \mathbb {Q} _{p}}ifp> 2. Also,−1is the only non-trivial torsion element inQ2{\displaystyle \mathbb {Q} _{2}}.
Given anatural numberk, theindexof the multiplicative group of thek-th powers of the non-zero elements ofQp{\displaystyle \mathbb {Q} _{p}}inQp×{\displaystyle \mathbb {Q} _{p}^{\times }}is finite.
The numbere, defined as the sum ofreciprocalsoffactorials, is not a member of anyp-adic field; butep∈Qp{\displaystyle e^{p}\in \mathbb {Q} _{p}}forp≠2{\displaystyle p\neq 2}. Forp= 2one must take at least the fourth power.[13](Thus a number with similar properties ase— namely ap-th root ofep— is a member ofQp{\displaystyle \mathbb {Q} _{p}}for allp.)
Helmut Hasse'slocal–global principleis said to hold for an equation if it can be solved over the rational numbersif and only ifit can be solved over the real numbers and over thep-adic numbers for every primep. This principle holds, for example, for equations given byquadratic forms, but fails for higher polynomials in several indeterminates.
The reals and thep-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance generalalgebraic number fields, in an analogous way. This will be described now.
SupposeDis aDedekind domainandEis itsfield of fractions. Pick a non-zeroprime idealPofD. Ifxis a non-zero element ofE, thenxDis afractional idealand can be uniquely factored as a product of positive and negative powers of non-zero prime ideals ofD. We write ordP(x) for the exponent ofPin this factorization, and for any choice of numbercgreater than 1 we can set
Completing with respect to this absolute value|⋅|Pyields a fieldEP, the proper generalization of the field ofp-adic numbers to this setting. The choice ofcdoes not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when theresidue fieldD/Pis finite, to take forcthe size ofD/P.
For example, whenEis anumber field,Ostrowski's theoremsays that every non-trivialnon-Archimedean absolute valueonEarises as some|⋅|P. The remaining non-trivial absolute values onEarise from the different embeddings ofEinto the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings ofEinto the fieldsCp, thus putting the description of all
the non-trivial absolute values of a number field on a common footing.)
Often, one needs to simultaneously keep track of all the above-mentioned completions whenEis a number field (or more generally aglobal field), which are seen as encoding "local" information. This is accomplished byadele ringsandidele groups.
p-adic integers can be extended top-adic solenoidsTp{\displaystyle \mathbb {T} _{p}}. There is a map fromTp{\displaystyle \mathbb {T} _{p}}to thecircle groupwhose fibers are thep-adic integersZp{\displaystyle \mathbb {Z} _{p}}, in analogy to how there is a map fromR{\displaystyle \mathbb {R} }to the circle whose fibers areZ{\displaystyle \mathbb {Z} }.
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Inmathematics, theinverse limit(also called theprojective limit) is a construction that allows one to "glue together" several relatedobjects, the precise gluing process being specified bymorphismsbetween the objects. Thus, inverse limits can be defined in anycategoryalthough their existence depends on the category that is considered. They are a special case of the concept oflimitin category theory.
By working in thedual category, that is by reversing the arrows, an inverse limit becomes adirect limitorinductive limit, and alimitbecomes acolimit.
We start with the definition of aninverse system(or projective system) ofgroupsandhomomorphisms. Let(I,≤){\displaystyle (I,\leq )}be adirectedposet(not all authors requireIto be directed). Let (Ai)i∈Ibe afamilyof groups and suppose we have a family of homomorphismsfij:Aj→Ai{\displaystyle f_{ij}:A_{j}\to A_{i}}for alli≤j{\displaystyle i\leq j}(note the order) with the following properties:
Then the pair((Ai)i∈I,(fij)i≤j∈I){\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}is called an inverse system of groups and morphisms overI{\displaystyle I}, and the morphismsfij{\displaystyle f_{ij}}are called the transition morphisms of the system.
We define theinverse limitof the inverse system((Ai)i∈I,(fij)i≤j∈I){\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}as a particularsubgroupof thedirect productof theAi{\displaystyle A_{i}}'s:
The inverse limitA{\displaystyle A}comes equipped withnatural projectionsπi:A→Aiwhich pick out theith component of the direct product for eachi{\displaystyle i}inI{\displaystyle I}. The inverse limit and the natural projections satisfy auniversal propertydescribed in the next section.
This same construction may be carried out if theAi{\displaystyle A_{i}}'s aresets,[1]semigroups,[1]topological spaces,[1]rings,modules(over a fixed ring),algebras(over a fixed ring), etc., and thehomomorphismsare morphisms in the correspondingcategory. The inverse limit will also belong to that category.
The inverse limit can be defined abstractly in an arbitrarycategoryby means of auniversal property. Let(Xi,fij){\textstyle (X_{i},f_{ij})}be an inverse system of objects andmorphismsin a categoryC(same definition as above). Theinverse limitof this system is an objectXinCtogether with morphismsπi:X→Xi(calledprojections) satisfyingπi=fij{\displaystyle f_{ij}}∘πjfor alli≤j. The pair (X,πi) must be universal in the sense that for any other such pair (Y, ψi) there exists a unique morphismu:Y→Xsuch that the diagram
commutesfor alli≤j. The inverse limit is often denoted
with the inverse system(Xi,fij){\textstyle (X_{i},f_{ij})}and the canonical projectionsπi{\displaystyle \pi _{i}}being understood.
In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limitsXandX'of an inverse system, there exists auniqueisomorphismX′ →Xcommuting with the projection maps.
Inverse systems and inverse limits in a categoryCadmit an alternative description in terms offunctors. Any partially ordered setIcan be considered as asmall categorywhere the morphisms consist of arrowsi→jif and only ifi≤j. An inverse system is then just acontravariant functorI→C. LetCIop{\displaystyle C^{I^{\mathrm {op} }}}be the category of these functors (withnatural transformationsas morphisms). An objectXofCcan be considered a trivial inverse system, where all objects are equal toXand all arrow are the identity ofX. This defines a "trivial functor" fromCtoCIop.{\displaystyle C^{I^{\mathrm {op} }}.}The inverse limit, if it exists, is defined as aright adjointof this trivial functor.
For anabelian categoryC, the inverse limit functor
isleft exact. IfIis ordered (not simply partially ordered) andcountable, andCis the categoryAbof abelian groups, the Mittag-Leffler condition is a condition on the transition morphismsfijthat ensures the exactness oflim←{\displaystyle \varprojlim }. Specifically,Eilenbergconstructed a functor
(pronounced "lim one") such that if (Ai,fij), (Bi,gij), and (Ci,hij) are three inverse systems of abelian groups, and
is ashort exact sequenceof inverse systems, then
is an exact sequence inAb.
If the ranges of the morphisms of an inverse system of abelian groups (Ai,fij) arestationary, that is, for everykthere existsj≥ksuch that for alli≥j:fkj(Aj)=fki(Ai){\displaystyle f_{kj}(A_{j})=f_{ki}(A_{i})}one says that the system satisfies theMittag-Leffler condition.
The name "Mittag-Leffler" for this condition was given by Bourbaki in their chapter on uniform structures for a similar result about inverse limits of complete Hausdorff uniform spaces. Mittag-Leffler used a similar argument in the proof ofMittag-Leffler's theorem.
The following situations are examples where the Mittag-Leffler condition is satisfied:
An example wherelim←1{\displaystyle \varprojlim {}^{1}}is non-zero is obtained by takingIto be the non-negativeintegers, lettingAi=piZ,Bi=Z, andCi=Bi/Ai=Z/piZ. Then
whereZpdenotes thep-adic integers.
More generally, ifCis an arbitrary abelian category that hasenough injectives, then so doesCI, and the rightderived functorsof the inverse limit functor can thus be defined. Thenth right derived functor is denoted
In the case whereCsatisfiesGrothendieck's axiom(AB4*),Jan-Erik Roosgeneralized the functor lim1onAbIto series of functors limnsuch that
It was thought for almost 40 years that Roos had proved (inSur les foncteurs dérivés de lim. Applications.) that lim1Ai= 0 for (Ai,fij) an inverse system with surjective transition morphisms andIthe set of non-negative integers (such inverse systems are often called "Mittag-Lefflersequences"). However, in 2002,Amnon NeemanandPierre Deligneconstructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim1Ai≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct ifChas a set of generators (in addition to satisfying (AB3) and (AB4*)).
Barry Mitchellhas shown (in "The cohomological dimension of a directed set") that ifIhascardinalityℵd{\displaystyle \aleph _{d}}(thedthinfinite cardinal), thenRnlim is zero for alln≥d+ 2. This applies to theI-indexed diagrams in the category ofR-modules, withRa commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which limn, on diagrams indexed by a countable set, is nonzero forn> 1).
Thecategorical dualof an inverse limit is adirect limit(or inductive limit). More general concepts are thelimits and colimitsof category theory. The terminology is somewhat confusing: inverse limits are a class of limits, while direct limits are a class of colimits.
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In mathematics, adyadic rationalorbinary rationalis a number that can be expressed as afractionwhosedenominatoris apower of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important incomputer sciencebecause they are the only ones with finitebinary representations. Dyadic rationals also have applications in weights and measures, musicaltime signatures, and early mathematics education. They can accurately approximate anyreal number.
The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form aring, lying between the ring ofintegersand thefieldofrational numbers. This ring may be denotedZ[12]{\displaystyle \mathbb {Z} [{\tfrac {1}{2}}]}.
In advanced mathematics, the dyadic rational numbers are central to the constructions of thedyadic solenoid,Minkowski's question-mark function,Daubechies wavelets,Thompson's group,Prüfer 2-group,surreal numbers, andfusible numbers. These numbers areorder-isomorphicto the rational numbers; they form a subsystem of the2-adic numbersas well as of the reals, and can represent thefractional partsof 2-adic numbers. Functions from natural numbers to dyadic rationals have been used to formalizemathematical analysisinreverse mathematics.
Many traditional systems of weights and measures are based on the idea of repeated halving, which produces dyadic rationals when measuring fractional amounts of units. Theinchis customarily subdivided in dyadic rationals rather than using a decimal subdivision.[1]The customary divisions of thegalloninto half-gallons,quarts,pints, andcupsare also dyadic.[2]The ancient Egyptians used dyadic rationals in measurement, with denominators up to 64.[3]Similarly, systems of weights from theIndus Valley civilisationare for the most part based on repeated halving; anthropologist Heather M.-L. Miller writes that "halving is a relatively simple operation with beam balances, which is likely why so many weight systems of this time period used binary systems".[4]
Dyadic rationals are central tocomputer scienceas a type of fractional number that many computers can manipulate directly.[5]In particular, as a data type used by computers,floating-point numbersare often defined as integers multiplied by positive or negative powers of two. The numbers that can be represented precisely in a floating-point format, such as theIEEE floating-point datatypes, are called its representable numbers. For most floating-point representations, the representable numbers are a subset of the dyadic rationals.[6]The same is true forfixed-point datatypes, which also use powers of two implicitly in the majority of cases.[7]Because of the simplicity of computing with dyadic rationals, they are also used for exact real computing usinginterval arithmetic,[8]and are central to some theoretical models ofcomputable numbers.[9][10][11]
Generating arandom variablefrom random bits, in a fixed amount of time, is possible only when the variable has finitely many outcomes whose probabilities are all dyadic rational numbers. For random variables whose probabilities are not dyadic, it is necessary either to approximate their probabilities by dyadic rationals, or to use a random generation process whose time is itself random and unbounded.[12]
Time signaturesin Westernmusical notationtraditionally are written in a form resembling fractions (for example:22,44, or68),[13]although the horizontal line of the musical staff that separates the top and bottom number is usually omitted when writing the signature separately from its staff. As fractions they are generally dyadic,[14]althoughnon-dyadic time signatureshave also been used.[15]The numeric value of the signature, interpreted as a fraction, describes the length of a measure as a fraction of awhole note. Its numerator describes the number of beats per measure, and the denominator describes the length of each beat.[13][14]
In theories of childhood development of the concept of a fraction based on the work ofJean Piaget, fractional numbers arising from halving and repeated halving are among the earliest forms of fractions to develop.[16]This stage of development of the concept of fractions has been called "algorithmic halving".[17]Addition and subtraction of these numbers can be performed in steps that only involve doubling, halving, adding, and subtracting integers. In contrast, addition and subtraction of more general fractions involves integer multiplication and factorization to reach a common denominator. Therefore, dyadic fractions can be easier for students to calculate with than more general fractions.[18]
The dyadic numbers are therational numbersthat result from dividing anintegerby apower of two.[9]A rational numberp/q{\displaystyle p/q}in simplest terms is a dyadic rational whenq{\displaystyle q}is a power of two.[19]Another equivalent way of defining the dyadic rationals is that they are thereal numbersthat have a terminatingbinary representation.[9]
Addition,subtraction, andmultiplicationof any two dyadic rationals produces another dyadic rational, according to the following formulas:[20]
However, the result ofdividingone dyadic rational by another is not necessarily a dyadic rational.[21]For instance, 1 and 3 are both dyadic rational numbers, but 1/3 is not.
Every integer, and everyhalf-integer, is a dyadic rational.[22]They both meet the definition of being an integer divided by a power of two: every integer is an integer divided by one (the zeroth power of two), and every half-integer is an integer divided by two.
Everyreal numbercan be arbitrarily closely approximated by dyadic rationals. In particular, for a real numberx{\displaystyle x}, consider the dyadic rationals of the form⌊2ix⌋/2i{\textstyle \lfloor 2^{i}x\rfloor /2^{i}},wherei{\displaystyle i}can be any integer and⌊…⌋{\displaystyle \lfloor \dots \rfloor }denotes thefloor functionthat rounds its argument down to an integer. These numbers approximatex{\displaystyle x}from below to within an error of1/2i{\displaystyle 1/2^{i}}, which can be made arbitrarily small by choosingi{\displaystyle i}to be arbitrarily large. For afractalsubset of the real numbers, this error bound is within a constant factor of optimal: for these numbers, there is no approximationn/2i{\displaystyle n/2^{i}}with error smaller than a constant times1/2i{\displaystyle 1/2^{i}}.[23][24]The existence of accurate dyadic approximations can be expressed by saying that the set of all dyadic rationals isdensein thereal line.[22]More strongly, this set is uniformly dense, in the sense that the dyadic rationals with denominator2i{\displaystyle 2^{i}}are uniformly spaced on the real line.[9]
The dyadic rationals are precisely those numbers possessing finitebinary expansions.[9]Their binary expansions are not unique; there is one finite and one infinite representation of each dyadic rational other than 0 (ignoring terminal 0s). For example, 0.112= 0.10111...2, giving two different representations for 3/4.[9][25]The dyadic rationals are the only numbers whose binary expansions are not unique.[9]
Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic rationals are aringbut not afield.[26]The ring of dyadic rationals may be denotedZ[12]{\displaystyle \mathbb {Z} [{\tfrac {1}{2}}]}, meaning that it can be generated by evaluating polynomials with integer coefficients, at the argument 1/2.[27]As a ring, the dyadic rationals are asubringof the rational numbers, and anoverringof the integers.[28]Algebraically, this ring is thelocalizationof the integers with respect to the set ofpowers of two.[29]
As well as forming a subring of thereal numbers, the dyadic rational numbers form a subring of the2-adic numbers, a system of numbers that can be defined from binary representations that are finite to the right of the binary point but may extend infinitely far to the left. The 2-adic numbers include all rational numbers, not just the dyadic rationals. Embedding the dyadic rationals into the 2-adic numbers does not change the arithmetic of the dyadic rationals, but it gives them a different topological structure than they have as a subring of the real numbers. As they do in the reals, the dyadic rationals form a dense subset of the 2-adic numbers,[30]and are the set of 2-adic numbers with finite binary expansions. Every 2-adic number can be decomposed into the sum of a 2-adic integer and a dyadic rational; in this sense, the dyadic rationals can represent thefractional partsof 2-adic numbers, but this decomposition is not unique.[31]
Addition of dyadic rationals modulo 1 (thequotient groupZ[12]/Z{\displaystyle \mathbb {Z} [{\tfrac {1}{2}}]/\mathbb {Z} }of the dyadic rationals by the integers) forms thePrüfer 2-group.[32]
Considering only the addition and subtraction operations of the dyadic rationals gives them the structure of an additiveabelian group.Pontryagin dualityis a method for understanding abelian groups by constructing dual groups, whose elements arecharactersof the original group,group homomorphismsto the multiplicative group of thecomplex numbers, with pointwise multiplication as the dual group operation. The dual group of the additive dyadic rationals, constructed in this way, can also be viewed as atopological group. It is called the dyadic solenoid, and is isomorphic to the topological product of the real numbers and 2-adic numbers,quotientedby thediagonal embeddingof the dyadic rationals into this product.[30]It is an example of aprotorus, asolenoid, and anindecomposable continuum.[33]
Because they are a dense subset of the real numbers, the dyadic rationals, with their numeric ordering, form adense order. As with any two unbounded countable dense linear orders, byCantor's isomorphism theorem,[34]the dyadic rationals areorder-isomorphicto the rational numbers. In this case,Minkowski's question-mark functionprovides an order-preservingbijectionbetween the set of all rational numbers and the set of dyadic rationals.[35]
The dyadic rationals play a key role in the analysis ofDaubechies wavelets, as the set of points where thescaling functionof these wavelets is non-smooth.[26]Similarly, the dyadic rationals parameterize the discontinuities in the boundary between stable and unstable points in the parameter space of theHénon map.[36]
The set ofpiecewise linearhomeomorphismsfrom theunit intervalto itself that have power-of-2 slopes and dyadic-rational breakpoints forms a group under the operation offunction composition. This isThompson's group, the first known example of an infinite butfinitely presentedsimple group.[37]The same group can also be represented by an action on rooted binary trees,[38]or by an action on the dyadic rationals within the unit interval.[32]
Inreverse mathematics, one way of constructing thereal numbersis to represent them as functions fromunary numbersto dyadic rationals, where the value of one of these functions for the argumenti{\displaystyle i}is a dyadic rational with denominator2i{\displaystyle 2^{i}}that approximates the given real number. Defining real numbers in this way allows many of the basic results ofmathematical analysisto be proven within a restricted theory ofsecond-order arithmeticcalled "feasible analysis" (BTFA).[39]
Thesurreal numbersare generated by an iterated construction principle which starts by generating all finite dyadic rationals, and then goes on to create new and strange kinds of infinite, infinitesimal and other numbers.[40]This number system is foundational tocombinatorial game theory, and dyadic rationals arise naturally in this theory as the set of values of certain combinatorial games.[41][42][19]
Thefusible numbersare a subset of the dyadic rationals, the closure of the set{0}{\displaystyle \{0\}}under the operationx,y↦(x+y+1)/2{\displaystyle x,y\mapsto (x+y+1)/2}, restricted to pairsx,y{\displaystyle x,y}with|x−y|<1{\displaystyle |x-y|<1}. They arewell-ordered, withorder typeequal to theepsilon numberε0{\displaystyle \varepsilon _{0}}. For each integern{\displaystyle n}the smallest fusible number that is greater thann{\displaystyle n}has the formn+1/2k{\displaystyle n+1/2^{k}}. The existence ofk{\displaystyle k}for eachn{\displaystyle n}cannot be proven inPeano arithmetic,[43]andk{\displaystyle k}grows so rapidly as a function ofn{\displaystyle n}that forn=3{\displaystyle n=3}it is (inKnuth's up-arrow notationfor large numbers) already larger than2↑916{\displaystyle 2\uparrow ^{9}16}.[44]
The usual proof ofUrysohn's lemmautilizes the dyadic fractions for constructing the separating function from the lemma.
Ring homomorphisms
Algebraic structures
Related structures
Algebraic number theory
Noncommutative algebraic geometry
Free algebra
Clifford algebra
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Inmathematics, particularlyset theory, afinite setis asetthat has afinitenumber ofelements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
is a finite set with five elements. The number of elements of a finite set is anatural number(possibly zero) and is called thecardinality(or thecardinal number)of the set. A set that is not a finite set is called aninfinite set. For example, the set of all positive integers is infinite:
Finite sets are particularly important incombinatorics, the mathematical study ofcounting. Many arguments involving finite sets rely on thepigeonhole principle, which states that there cannot exist aninjectivefunctionfrom a larger finite set to a smaller finite set.
Formally, a setS{\displaystyle S}is calledfiniteif there exists abijection
for some natural numbern{\displaystyle n}(natural numbers are defined as sets inZermelo-Fraenkel set theory). The numbern{\displaystyle n}is the set's cardinality, denoted as|S|{\displaystyle |S|}.
If a nonempty set is finite, its elements may be written in asequence:
Ifn≥2, then there are multiple such sequences.
Incombinatorics, a finite set withn{\displaystyle n}elements is sometimes called ann{\displaystyle n}-setand asubsetwithk{\displaystyle k}elements is called ak{\displaystyle k}-subset. For example, the set{5,6,7}{\displaystyle \{5,6,7\}}is a 3-set – a finite set with three elements – and{6,7}{\displaystyle \{6,7\}}is a 2-subset of it.
Anyproper subsetof a finite setS{\displaystyle S}is finite and has fewer elements thanSitself. As a consequence, there cannot exist abijectionbetween a finite setSand a proper subset ofS. Any set with this property is calledDedekind-finite. Using the standardZFCaxioms forset theory, every Dedekind-finite set is also finite, but this implication cannot beprovedin ZF (Zermelo–Fraenkel axioms without theaxiom of choice) alone.
Theaxiom of countable choice, a weak version of the axiom of choice, is sufficient to prove this equivalence.
Any injective function between two finite sets of the same cardinality is also asurjective function(a surjection). Similarly, any surjection between two finite sets of the same cardinality is also an injection.
Theunionof two finite sets is finite, with
In fact, by theinclusion–exclusion principle:
More generally, the union of any finite number of finite sets is finite. TheCartesian productof finite sets is also finite, with:
Similarly, the Cartesian product of finitely many finite sets is finite. A finite set withn{\displaystyle n}elements has2n{\displaystyle 2^{n}}distinct subsets. That is, thepower set℘(S){\displaystyle \wp (S)}of a finite setSis finite, with cardinality2|S|{\displaystyle 2^{|S|}}.
Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite.
All finite sets arecountable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.)
Thefree semilatticeover a finite set is the set of its non-empty subsets, with thejoin operationbeing given by set union.
InZermelo–Fraenkel set theorywithout the axiom of choice (ZF), the following conditions are all equivalent:[1]
If theaxiom of choiceis also assumed (theaxiom of countable choiceis sufficient),[4]then the following conditions are all equivalent:
In ZF set theory without theaxiom of choice, the following concepts of finiteness for a setS{\displaystyle S}are distinct. They are arranged in strictly decreasing order of strength, i.e. if a setS{\displaystyle S}meets a criterion in the list then it meets all of the following criteria. In the absence of the axiom of choice the reverse implications are all unprovable, but if the axiom of choice is assumed then all of these concepts are equivalent.[5](Note that none of these definitions need the set of finiteordinal numbersto be defined first; they are all pure "set-theoretic" definitions in terms of the equality and membership relations, not involving ω.)
The forward implications (from strong to weak) are theorems within ZF. Counter-examples to the reverse implications (from weak to strong) in ZF withurelementsare found usingmodel theory.[7]
Most of these finiteness definitions and their names are attributed toTarski 1954byHoward & Rubin 1998, p. 278. However, definitions I, II, III, IV and V were presented inTarski 1924, pp. 49, 93, together with proofs (or references to proofs) for the forward implications. At that time, model theory was not sufficiently advanced to find the counter-examples.
Each of the properties I-finite thru IV-finite is a notion of smallness in the sense that any subset of a set with such a property will also have the property. This is not true for V-finite thru VII-finite because they may have countably infinite subsets.
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This article listsmathematicalproperties and laws ofsets, involving the set-theoreticoperationsofunion,intersection, andcomplementationand therelationsof setequalityand setinclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Thebinary operationsof set union (∪{\displaystyle \cup }) and intersection (∩{\displaystyle \cap }) satisfy many identities. Several of these identities or "laws" have well established names.
Throughout this article, capital letters (such asA,B,C,L,M,R,S,{\displaystyle A,B,C,L,M,R,S,}andX{\displaystyle X}) will denote sets. On the left hand side of an identity, typically,
This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity.[note 1]For example, the identity(L∖M)∖R=(L∖R)∖(M∖R){\displaystyle (L\,\setminus \,M)\,\setminus \,R~=~(L\,\setminus \,R)\,\setminus \,(M\,\setminus \,R)}may be read as:(Left set∖Middle set)∖Right set=(Left set∖Right set)∖(Middle set∖Right set).{\displaystyle ({\text{Left set}}\,\setminus \,{\text{Middle set}})\,\setminus \,{\text{Right set}}~=~({\text{Left set}}\,\setminus \,{\text{Right set}})\,\setminus \,({\text{Middle set}}\,\setminus \,{\text{Right set}}).}
For setsL{\displaystyle L}andR,{\displaystyle R,}define:L∪R=def{x:x∈Lorx∈R}L∩R=def{x:x∈Landx∈R}L∖R=def{x:x∈Landx∉R}{\displaystyle {\begin{alignedat}{4}L\cup R&&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x\in L\;&&{\text{ or }}\;\,&&\;x\in R~\}\\L\cap R&&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x\in L\;&&{\text{ and }}&&\;x\in R~\}\\L\setminus R&&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x\in L\;&&{\text{ and }}&&\;x\notin R~\}\\\end{alignedat}}}andL△R=def{x:xbelongs to exactly one ofLandR}{\displaystyle L\triangle R~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x{\text{ belongs to exactly one of }}L{\text{ and }}R~\}}where thesymmetric differenceL△R{\displaystyle L\triangle R}is sometimes denoted byL⊖R{\displaystyle L\ominus R}and equals:[1][2]L△R=(L∖R)∪(R∖L)=(L∪R)∖(L∩R).{\displaystyle {\begin{alignedat}{4}L\;\triangle \;R~&=~(L~\setminus ~&&R)~\cup ~&&(R~\setminus ~&&L)\\~&=~(L~\cup ~&&R)~\setminus ~&&(L~\cap ~&&R).\end{alignedat}}}
One setL{\displaystyle L}is said tointersectanother setR{\displaystyle R}ifL∩R≠∅.{\displaystyle L\cap R\neq \varnothing .}Sets that do not intersect are said to bedisjoint.
Thepower setofX{\displaystyle X}is the set of all subsets ofX{\displaystyle X}and will be denoted by℘(X)=def{L:L⊆X}.{\displaystyle \wp (X)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~L~:~L\subseteq X~\}.}
Universe set and complement notation
The notationL∁=defX∖L.{\displaystyle L^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus L.}may be used ifL{\displaystyle L}is a subset of some setX{\displaystyle X}that is understood (say from context, or because it is clearly stated what the supersetX{\displaystyle X}is).
It is emphasized that the definition ofL∁{\displaystyle L^{\complement }}depends on context. For instance, hadL{\displaystyle L}been declared as a subset ofY,{\displaystyle Y,}with the setsY{\displaystyle Y}andX{\displaystyle X}not necessarily related to each other in any way, thenL∁{\displaystyle L^{\complement }}would likely meanY∖L{\displaystyle Y\setminus L}instead ofX∖L.{\displaystyle X\setminus L.}
If it is needed then unless indicated otherwise, it should be assumed thatX{\displaystyle X}denotes theuniverse set, which means that all sets that are used in the formula are subsets ofX.{\displaystyle X.}In particular, thecomplement of a setL{\displaystyle L}will be denoted byL∁{\displaystyle L^{\complement }}where unless indicated otherwise, it should be assumed thatL∁{\displaystyle L^{\complement }}denotes the complement ofL{\displaystyle L}in (the universe)X.{\displaystyle X.}
AssumeL⊆X.{\displaystyle L\subseteq X.}
Identity:[3]
Definition:e{\displaystyle e}is called aleft identity elementof abinary operator∗{\displaystyle \,\ast \,}ife∗R=R{\displaystyle e\,\ast \,R=R}for allR{\displaystyle R}and it is called aright identity elementof∗{\displaystyle \,\ast \,}ifL∗e=L{\displaystyle L\,\ast \,e=L}for allL.{\displaystyle L.}A left identity element that is also a right identity element if called anidentity element.
The empty set∅{\displaystyle \varnothing }is an identity element of binary union∪{\displaystyle \cup }and symmetric difference△,{\displaystyle \triangle ,}and it is also a right identity element of set subtraction∖:{\displaystyle \,\setminus :}
L∩X=L=X∩LwhereL⊆XL∪∅=L=∅∪LL△∅=L=∅△LL∖∅=L{\displaystyle {\begin{alignedat}{10}L\cap X&\;=\;&&L&\;=\;&X\cap L~~~~{\text{ where }}L\subseteq X\\[1.4ex]L\cup \varnothing &\;=\;&&L&\;=\;&\varnothing \cup L\\[1.4ex]L\,\triangle \varnothing &\;=\;&&L&\;=\;&\varnothing \,\triangle L\\[1.4ex]L\setminus \varnothing &\;=\;&&L\\[1.4ex]\end{alignedat}}}but∅{\displaystyle \varnothing }is not a left identity element of∖{\displaystyle \,\setminus \,}since∅∖L=∅{\displaystyle \varnothing \setminus L=\varnothing }so∅∖L=L{\textstyle \varnothing \setminus L=L}if and only ifL=∅.{\displaystyle L=\varnothing .}
Idempotence[3]L∗L=L{\displaystyle L\ast L=L}andNilpotenceL∗L=∅{\displaystyle L\ast L=\varnothing }:
L∪L=L(Idempotence)L∩L=L(Idempotence)L△L=∅(Nilpotence of index 2)L∖L=∅(Nilpotence of index 2){\displaystyle {\begin{alignedat}{10}L\cup L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\cap L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\,\triangle \,L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]L\setminus L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]\end{alignedat}}}
Domination[3]/Absorbing element:
Definition:z{\displaystyle z}is called aleft absorbing elementof abinary operator∗{\displaystyle \,\ast \,}ifz∗R=z{\displaystyle z\,\ast \,R=z}for allR{\displaystyle R}and it is called aright absorbing elementof∗{\displaystyle \,\ast \,}ifL∗z=z{\displaystyle L\,\ast \,z=z}for allL.{\displaystyle L.}A left absorbing element that is also a right absorbing element if called anabsorbing element. Absorbing elements are also sometime calledannihilating elementsorzero elements.
A universe set is an absorbing element of binary union∪.{\displaystyle \cup .}The empty set∅{\displaystyle \varnothing }is an absorbing element of binary intersection∩{\displaystyle \cap }and binary Cartesian product×,{\displaystyle \times ,}and it is also a left absorbing element of set subtraction∖:{\displaystyle \,\setminus :}
X∪L=X=L∪XwhereL⊆X∅∩L=∅=L∩∅∅×L=∅=L×∅∅∖L=∅{\displaystyle {\begin{alignedat}{10}X\cup L&\;=\;&&X&\;=\;&L\cup X~~~~{\text{ where }}L\subseteq X\\[1.4ex]\varnothing \cap L&\;=\;&&\varnothing &\;=\;&L\cap \varnothing \\[1.4ex]\varnothing \times L&\;=\;&&\varnothing &\;=\;&L\times \varnothing \\[1.4ex]\varnothing \setminus L&\;=\;&&\varnothing &\;\;&\\[1.4ex]\end{alignedat}}}but∅{\displaystyle \varnothing }is not a right absorbing element of set subtraction sinceL∖∅=L{\displaystyle L\setminus \varnothing =L}whereL∖∅=∅{\textstyle L\setminus \varnothing =\varnothing }if and only ifL=∅.{\textstyle L=\varnothing .}
Double complementorinvolutionlaw:
X∖(X∖L)=LAlso written(L∁)∁=LwhereL⊆X(Double complement/Involution law){\displaystyle {\begin{alignedat}{10}X\setminus (X\setminus L)&=L&&\qquad {\text{ Also written }}\quad &&\left(L^{\complement }\right)^{\complement }=L&&\quad &&{\text{ where }}L\subseteq X\quad {\text{ (Double complement/Involution law)}}\\[1.4ex]\end{alignedat}}}
L∖∅=L{\displaystyle L\setminus \varnothing =L}∅=L∖L=∅∖L=L∖XwhereL⊆X{\displaystyle {\begin{alignedat}{4}\varnothing &=L&&\setminus L\\&=\varnothing &&\setminus L\\&=L&&\setminus X~~~~{\text{ where }}L\subseteq X\\\end{alignedat}}}[3]
L∁=X∖L(definition of notation){\displaystyle L^{\complement }=X\setminus L\quad {\text{ (definition of notation)}}}
L∪(X∖L)=XAlso writtenL∪L∁=XwhereL⊆XL△(X∖L)=XAlso writtenL△L∁=XwhereL⊆XL∩(X∖L)=∅Also writtenL∩L∁=∅{\displaystyle {\begin{alignedat}{10}L\,\cup (X\setminus L)&=X&&\qquad {\text{ Also written }}\quad &&L\cup L^{\complement }=X&&\quad &&{\text{ where }}L\subseteq X\\[1.4ex]L\,\triangle (X\setminus L)&=X&&\qquad {\text{ Also written }}\quad &&L\,\triangle L^{\complement }=X&&\quad &&{\text{ where }}L\subseteq X\\[1.4ex]L\,\cap (X\setminus L)&=\varnothing &&\qquad {\text{ Also written }}\quad &&L\cap L^{\complement }=\varnothing &&\quad &&\\[1.4ex]\end{alignedat}}}[3]
X∖∅=XAlso written∅∁=X(Complement laws for the empty set))X∖X=∅Also writtenX∁=∅(Complement laws for the universe set){\displaystyle {\begin{alignedat}{10}X\setminus \varnothing &=X&&\qquad {\text{ Also written }}\quad &&\varnothing ^{\complement }=X&&\quad &&{\text{ (Complement laws for the empty set))}}\\[1.4ex]X\setminus X&=\varnothing &&\qquad {\text{ Also written }}\quad &&X^{\complement }=\varnothing &&\quad &&{\text{ (Complement laws for the universe set)}}\\[1.4ex]\end{alignedat}}}
In the left hand sides of the following identities,L{\displaystyle L}is theLeft most set andR{\displaystyle R}is theRight most set.
Assume bothLandR{\displaystyle L{\text{ and }}R}are subsets of some universe setX.{\displaystyle X.}
In the left hand sides of the following identities,Lis theLeft most set andRis theRight most set. Whenever necessary, bothLandRshould be assumed to be subsets of some universe setX, so thatL∁:=X∖LandR∁:=X∖R.{\displaystyle L^{\complement }:=X\setminus L{\text{ and }}R^{\complement }:=X\setminus R.}
L∩R=L∖(L∖R)=R∖(R∖L)=L∖(L△R)=L△(L∖R){\displaystyle {\begin{alignedat}{9}L\cap R&=L&&\,\,\setminus \,&&(L&&\,\,\setminus &&R)\\&=R&&\,\,\setminus \,&&(R&&\,\,\setminus &&L)\\&=L&&\,\,\setminus \,&&(L&&\,\triangle \,&&R)\\&=L&&\,\triangle \,&&(L&&\,\,\setminus &&R)\\\end{alignedat}}}
L∪R=(L△R)∪L=(L△R)△(L∩R)=(R∖L)∪L(union is disjoint){\displaystyle {\begin{alignedat}{9}L\cup R&=(&&L\,\triangle \,R)&&\,\,\cup &&&&L&&&&\\&=(&&L\,\triangle \,R)&&\,\triangle \,&&(&&L&&\cap \,&&R)\\&=(&&R\,\setminus \,L)&&\,\,\cup &&&&L&&&&~~~~~{\text{ (union is disjoint)}}\\\end{alignedat}}}
L△R=R△L=(L∪R)∖(L∩R)=(L∖R)∪(R∖L)(union is disjoint)=(L△M)△(M△R)whereMis an arbitrary set.=(L∁)△(R∁){\displaystyle {\begin{alignedat}{9}L\,\triangle \,R&=&&R\,\triangle \,L&&&&&&&&\\&=(&&L\,\cup \,R)&&\,\setminus \,&&(&&L\,\,\cap \,R)&&\\&=(&&L\,\setminus \,R)&&\cup \,&&(&&R\,\,\setminus \,L)&&~~~~~{\text{ (union is disjoint)}}\\&=(&&L\,\triangle \,M)&&\,\triangle \,&&(&&M\,\triangle \,R)&&~~~~~{\text{ where }}M{\text{ is an arbitrary set. }}\\&=(&&L^{\complement })&&\,\triangle \,&&(&&R^{\complement })&&\\\end{alignedat}}}
L∖R=L∖(L∩R)=L∩(L△R)=L△(L∩R)=R△(L∪R){\displaystyle {\begin{alignedat}{9}L\setminus R&=&&L&&\,\,\setminus &&(L&&\,\,\cap &&R)\\&=&&L&&\,\,\cap &&(L&&\,\triangle \,&&R)\\&=&&L&&\,\triangle \,&&(L&&\,\,\cap &&R)\\&=&&R&&\,\triangle \,&&(L&&\,\,\cup &&R)\\\end{alignedat}}}
De Morgan's lawsstate that forL,R⊆X:{\displaystyle L,R\subseteq X:}
X∖(L∩R)=(X∖L)∪(X∖R)Also written(L∩R)∁=L∁∪R∁(De Morgan's law)X∖(L∪R)=(X∖L)∩(X∖R)Also written(L∪R)∁=L∁∩R∁(De Morgan's law){\displaystyle {\begin{alignedat}{10}X\setminus (L\cap R)&=(X\setminus L)\cup (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cap R)^{\complement }=L^{\complement }\cup R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]X\setminus (L\cup R)&=(X\setminus L)\cap (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cup R)^{\complement }=L^{\complement }\cap R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]\end{alignedat}}}
Unions, intersection, and symmetric difference arecommutative operations:[3]
L∪R=R∪L(Commutativity)L∩R=R∩L(Commutativity)L△R=R△L(Commutativity){\displaystyle {\begin{alignedat}{10}L\cup R&\;=\;&&R\cup L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\cap R&\;=\;&&R\cap L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\,\triangle R&\;=\;&&R\,\triangle L&&\quad {\text{ (Commutativity)}}\\[1.4ex]\end{alignedat}}}
Set subtraction is not commutative. However, the commutativity of set subtraction can be characterized: from(L∖R)∩(R∖L)=∅{\displaystyle (L\,\setminus \,R)\cap (R\,\setminus \,L)=\varnothing }it follows that:L∖R=R∖Lif and only ifL=R.{\displaystyle L\,\setminus \,R=R\,\setminus \,L\quad {\text{ if and only if }}\quad L=R.}Said differently, if distinct symbols always represented distinct sets, then theonlytrue formulas of the form⋅∖⋅=⋅∖⋅{\displaystyle \,\cdot \,\,\setminus \,\,\cdot \,=\,\cdot \,\,\setminus \,\,\cdot \,}that could be written would be those involving a single symbol; that is, those of the form:S∖S=S∖S.{\displaystyle S\,\setminus \,S=S\,\setminus \,S.}But such formulas are necessarily true foreverybinary operation∗{\displaystyle \,\ast \,}(becausex∗x=x∗x{\displaystyle x\,\ast \,x=x\,\ast \,x}must hold by definition ofequality), and so in this sense, set subtraction is as diametrically opposite to being commutative as is possible for a binary operation.
Set subtraction is also neitherleft alternativenorright alternative; instead,(L∖L)∖R=L∖(L∖R){\displaystyle (L\setminus L)\setminus R=L\setminus (L\setminus R)}if and only ifL∩R=∅{\displaystyle L\cap R=\varnothing }if and only if(R∖L)∖L=R∖(L∖L).{\displaystyle (R\setminus L)\setminus L=R\setminus (L\setminus L).}Set subtraction isquasi-commutativeand satisfies theJordan identity.
Absorption laws:
L∪(L∩R)=L(Absorption)L∩(L∪R)=L(Absorption){\displaystyle {\begin{alignedat}{4}L\cup (L\cap R)&\;=\;&&L&&\quad {\text{ (Absorption)}}\\[1.4ex]L\cap (L\cup R)&\;=\;&&L&&\quad {\text{ (Absorption)}}\\[1.4ex]\end{alignedat}}}
Other properties
L∖R=L∩(X∖R)Also writtenL∖R=L∩R∁whereL,R⊆XX∖(L∖R)=(X∖L)∪RAlso written(L∖R)∁=L∁∪RwhereR⊆XL∖R=(X∖R)∖(X∖L)Also writtenL∖R=R∁∖L∁whereL,R⊆X{\displaystyle {\begin{alignedat}{10}L\setminus R&=L\cap (X\setminus R)&&\qquad {\text{ Also written }}\quad &&L\setminus R=L\cap R^{\complement }&&\quad &&{\text{ where }}L,R\subseteq X\\[1.4ex]X\setminus (L\setminus R)&=(X\setminus L)\cup R&&\qquad {\text{ Also written }}\quad &&(L\setminus R)^{\complement }=L^{\complement }\cup R&&\quad &&{\text{ where }}R\subseteq X\\[1.4ex]L\setminus R&=(X\setminus R)\setminus (X\setminus L)&&\qquad {\text{ Also written }}\quad &&L\setminus R=R^{\complement }\setminus L^{\complement }&&\quad &&{\text{ where }}L,R\subseteq X\\[1.4ex]\end{alignedat}}}
Intervals:
(a,b)∩(c,d)=(max{a,c},min{b,d}){\displaystyle (a,b)\cap (c,d)=(\max\{a,c\},\min\{b,d\})}[a,b)∩[c,d)=[max{a,c},min{b,d}){\displaystyle [a,b)\cap [c,d)=[\max\{a,c\},\min\{b,d\})}
The following statements are equivalent for anyL,R⊆X:{\displaystyle L,R\subseteq X:}[3]
The following statements are equivalent for anyL,R⊆X:{\displaystyle L,R\subseteq X:}
The following statements are equivalent:
A setL{\displaystyle L}isemptyif the sentence∀x(x∉L){\displaystyle \forall x(x\not \in L)}is true, where the notationx∉L{\displaystyle x\not \in L}is shorthand for¬(x∈L).{\displaystyle \lnot (x\in L).}
IfL{\displaystyle L}is any set then the following are equivalent:
IfL{\displaystyle L}is any set then the following are equivalent:
Given anyx,{\displaystyle x,}the following are equivalent:
Moreover,(L∖R)∩R=∅always holds.{\displaystyle (L\setminus R)\cap R=\varnothing \qquad {\text{ always holds}}.}
Inclusion is apartial order:
Explicitly, this means thatinclusion⊆,{\displaystyle \,\subseteq ,\,}which is abinary operation, has the following three properties:[3]
The following proposition says that for any setS,{\displaystyle S,}thepower setofS,{\displaystyle S,}ordered by inclusion, is abounded lattice, and hence together with the distributive and complement laws above, show that it is aBoolean algebra.
Existence of aleast elementand agreatest element:∅⊆L⊆X{\displaystyle \varnothing \subseteq L\subseteq X}
Joins/supremums exist:[3]L⊆L∪R{\displaystyle L\subseteq L\cup R}
The unionL∪R{\displaystyle L\cup R}is the join/supremum ofL{\displaystyle L}andR{\displaystyle R}with respect to⊆{\displaystyle \,\subseteq \,}because:
The intersectionL∩R{\displaystyle L\cap R}is the join/supremum ofL{\displaystyle L}andR{\displaystyle R}with respect to⊇.{\displaystyle \,\supseteq .\,}
Meets/infimums exist:[3]L∩R⊆L{\displaystyle L\cap R\subseteq L}
The intersectionL∩R{\displaystyle L\cap R}is the meet/infimum ofL{\displaystyle L}andR{\displaystyle R}with respect to⊆{\displaystyle \,\subseteq \,}because:
The unionL∪R{\displaystyle L\cup R}is the meet/infimum ofL{\displaystyle L}andR{\displaystyle R}with respect to⊇.{\displaystyle \,\supseteq .\,}
Other inclusion properties:
L∖R⊆L{\displaystyle L\setminus R\subseteq L}(L∖R)∩L=L∖R{\displaystyle (L\setminus R)\cap L=L\setminus R}
In the left hand sides of the following identities,L{\displaystyle L}is theLeft most set,M{\displaystyle M}is theMiddle set, andR{\displaystyle R}is theRight most set.
There is no universal agreement on theorder of precedenceof the basic set operators.
Nevertheless, many authors useprecedence rulesfor set operators, although these rules vary with the author.
One common convention is to associate intersectionL∩R={x:(x∈L)∧(x∈R)}{\displaystyle L\cap R=\{x:(x\in L)\land (x\in R)\}}withlogical conjunction (and)L∧R{\displaystyle L\land R}and associate unionL∪R={x:(x∈L)∨(x∈R)}{\displaystyle L\cup R=\{x:(x\in L)\lor (x\in R)\}}withlogical disjunction (or)L∨R,{\displaystyle L\lor R,}and then transfer theprecedence of these logical operators(where∧{\displaystyle \,\land \,}has precedence over∨{\displaystyle \,\lor \,}) to these set operators, thereby giving∩{\displaystyle \,\cap \,}precedence over∪.{\displaystyle \,\cup .\,}So for example,L∪M∩R{\displaystyle L\cup M\cap R}would meanL∪(M∩R){\displaystyle L\cup (M\cap R)}since it would be associated with the logical statementL∨M∧R=L∨(M∧R){\displaystyle L\lor M\land R~=~L\lor (M\land R)}and similarly,L∪M∩R∪Z{\displaystyle L\cup M\cap R\cup Z}would meanL∪(M∩R)∪Z{\displaystyle L\cup (M\cap R)\cup Z}since it would be associated withL∨M∧R∨Z=L∨(M∧R)∨Z.{\displaystyle L\lor M\land R\lor Z~=~L\lor (M\land R)\lor Z.}
Sometimes, set complement (subtraction)∖{\displaystyle \,\setminus \,}is also associated withlogical complement (not)¬,{\displaystyle \,\lnot ,\,}in which case it will have the highest precedence.
More specifically,L∖R={x:(x∈L)∧¬(x∈R)}{\displaystyle L\setminus R=\{x:(x\in L)\land \lnot (x\in R)\}}is rewrittenL∧¬R{\displaystyle L\land \lnot R}so that for example,L∪M∖R{\displaystyle L\cup M\setminus R}would meanL∪(M∖R){\displaystyle L\cup (M\setminus R)}since it would be rewritten as the logical statementL∨M∧¬R{\displaystyle L\lor M\land \lnot R}which is equal toL∨(M∧¬R).{\displaystyle L\lor (M\land \lnot R).}For another example, becauseL∧¬M∧R{\displaystyle L\land \lnot M\land R}meansL∧(¬M)∧R,{\displaystyle L\land (\lnot M)\land R,}which is equal to both(L∧(¬M))∧R{\displaystyle (L\land (\lnot M))\land R}andL∧((¬M)∧R)=L∧(R∧(¬M)){\displaystyle L\land ((\lnot M)\land R)~=~L\land (R\land (\lnot M))}(where(¬M)∧R{\displaystyle (\lnot M)\land R}was rewritten asR∧(¬M){\displaystyle R\land (\lnot M)}), the formulaL∖M∩R{\displaystyle L\setminus M\cap R}would refer to the set(L∖M)∩R=L∩(R∖M);{\displaystyle (L\setminus M)\cap R=L\cap (R\setminus M);}moreover, sinceL∧(¬M)∧R=(L∧R)∧¬M,{\displaystyle L\land (\lnot M)\land R=(L\land R)\land \lnot M,}this set is also equal to(L∩R)∖M{\displaystyle (L\cap R)\setminus M}(other set identities can similarly be deduced frompropositional calculusidentitiesin this way).
However, because set subtraction is not associative(L∖M)∖R≠L∖(M∖R),{\displaystyle (L\setminus M)\setminus R\neq L\setminus (M\setminus R),}a formula such asL∖M∖R{\displaystyle L\setminus M\setminus R}would be ambiguous; for this reason, among others, set subtraction is often not assigned any precedence at all.
Symmetric differenceL△R={x:(x∈L)⊕(x∈R)}{\displaystyle L\triangle R=\{x:(x\in L)\oplus (x\in R)\}}is sometimes associated withexclusive or (xor)L⊕R{\displaystyle L\oplus R}(also sometimes denoted by⊻{\displaystyle \,\veebar }), in which case if the order of precedence from highest to lowest is¬,⊕,∧,∨{\displaystyle \,\lnot ,\,\oplus ,\,\land ,\,\lor \,}then the order of precedence (from highest to lowest) for the set operators would be∖,△,∩,∪.{\displaystyle \,\setminus ,\,\triangle ,\,\cap ,\,\cup .}There is no universal agreement on the precedence of exclusive disjunction⊕{\displaystyle \,\oplus \,}with respect to the other logical connectives, which is why symmetric difference△{\displaystyle \,\triangle \,}is not often assigned a precedence.
Definition: Abinary operator∗{\displaystyle \,\ast \,}is calledassociativeif(L∗M)∗R=L∗(M∗R){\displaystyle (L\,\ast \,M)\,\ast \,R=L\,\ast \,(M\,\ast \,R)}always holds.
The following set operators are associative:[3]
(L∪M)∪R=L∪(M∪R)(L∩M)∩R=L∩(M∩R)(L△M)△R=L△(M△R){\displaystyle {\begin{alignedat}{5}(L\cup M)\cup R&\;=\;\;&&L\cup (M\cup R)\\[1.4ex](L\cap M)\cap R&\;=\;\;&&L\cap (M\cap R)\\[1.4ex](L\,\triangle M)\,\triangle R&\;=\;\;&&L\,\triangle (M\,\triangle R)\\[1.4ex]\end{alignedat}}}
For set subtraction, instead of associativity, only the following is always guaranteed:(L∖M)∖R⊆L∖(M∖R){\displaystyle (L\,\setminus \,M)\,\setminus \,R\;~~{\color {red}{\subseteq }}~~\;L\,\setminus \,(M\,\setminus \,R)}where equality holds if and only ifL∩R=∅{\displaystyle L\cap R=\varnothing }(this condition does not depend onM{\displaystyle M}). Thus(L∖M)∖R=L∖(M∖R){\textstyle \;(L\setminus M)\setminus R=L\setminus (M\setminus R)\;}if and only if(R∖M)∖L=R∖(M∖L),{\displaystyle \;(R\setminus M)\setminus L=R\setminus (M\setminus L),\;}where the only difference between the left and right hand side set equalities is that the locations ofLandR{\displaystyle L{\text{ and }}R}have been swapped.
Definition: If∗and∙{\displaystyle \ast {\text{ and }}\bullet }arebinary operatorsthen∗{\displaystyle \,\ast \,}left distributesover∙{\displaystyle \,\bullet \,}ifL∗(M∙R)=(L∗M)∙(L∗R)for allL,M,R{\displaystyle L\,\ast \,(M\,\bullet \,R)~=~(L\,\ast \,M)\,\bullet \,(L\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R}while∗{\displaystyle \,\ast \,}right distributesover∙{\displaystyle \,\bullet \,}if(L∙M)∗R=(L∗R)∙(M∗R)for allL,M,R.{\displaystyle (L\,\bullet \,M)\,\ast \,R~=~(L\,\ast \,R)\,\bullet \,(M\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R.}The operator∗{\displaystyle \,\ast \,}distributesover∙{\displaystyle \,\bullet \,}if it both left distributes and right distributes over∙.{\displaystyle \,\bullet \,.\,}In the definitions above, to transform one side to the other, the innermost operator (the operator inside the parentheses) becomes the outermost operator and the outermost operator becomes the innermost operator.
Right distributivity:[3]
(L∩M)∪R=(L∪R)∩(M∪R)(Right-distributivity of∪over∩)(L∪M)∪R=(L∪R)∪(M∪R)(Right-distributivity of∪over∪)(L∪M)∩R=(L∩R)∪(M∩R)(Right-distributivity of∩over∪)(L∩M)∩R=(L∩R)∩(M∩R)(Right-distributivity of∩over∩)(L△M)∩R=(L∩R)△(M∩R)(Right-distributivity of∩over△)(L∩M)×R=(L×R)∩(M×R)(Right-distributivity of×over∩)(L∪M)×R=(L×R)∪(M×R)(Right-distributivity of×over∪)(L∖M)×R=(L×R)∖(M×R)(Right-distributivity of×over∖)(L△M)×R=(L×R)△(M×R)(Right-distributivity of×over△)(L∪M)∖R=(L∖R)∪(M∖R)(Right-distributivity of∖over∪)(L∩M)∖R=(L∖R)∩(M∖R)(Right-distributivity of∖over∩)(L△M)∖R=(L∖R)△(M∖R)(Right-distributivity of∖over△)(L∖M)∖R=(L∖R)∖(M∖R)(Right-distributivity of∖over∖)=L∖(M∪R){\displaystyle {\begin{alignedat}{9}(L\,\cap \,M)\,\cup \,R~&~~=~~&&(L\,\cup \,R)\,&&\cap \,&&(M\,\cup \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cup \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\cup \,R~&~~=~~&&(L\,\cup \,R)\,&&\cup \,&&(M\,\cup \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cup \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,&&\cup \,&&(M\,\cap \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cap \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\cap \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,&&\cap \,&&(M\,\cap \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cap \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,&&\triangle \,&&(M\,\cap \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cap \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex](L\,\cap \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cap \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cup \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\setminus \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\setminus \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\triangle \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)\,&&\cup \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\cap \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)\,&&\cap \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)&&\,\triangle \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex](L\,\setminus \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)&&\,\setminus \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex]~&~~=~~&&~~\;~~\;~~\;~L&&\,\setminus \,&&(M\cup R)\\[1.4ex]\end{alignedat}}}
Left distributivity:[3]
L∪(M∩R)=(L∪M)∩(L∪R)(Left-distributivity of∪over∩)L∪(M∪R)=(L∪M)∪(L∪R)(Left-distributivity of∪over∪)L∩(M∪R)=(L∩M)∪(L∩R)(Left-distributivity of∩over∪)L∩(M∩R)=(L∩M)∩(L∩R)(Left-distributivity of∩over∩)L∩(M△R)=(L∩M)△(L∩R)(Left-distributivity of∩over△)L×(M∩R)=(L×M)∩(L×R)(Left-distributivity of×over∩)L×(M∪R)=(L×M)∪(L×R)(Left-distributivity of×over∪)L×(M∖R)=(L×M)∖(L×R)(Left-distributivity of×over∖)L×(M△R)=(L×M)△(L×R)(Left-distributivity of×over△){\displaystyle {\begin{alignedat}{5}L\cup (M\cap R)&\;=\;\;&&(L\cup M)\cap (L\cup R)\qquad &&{\text{ (Left-distributivity of }}\,\cup \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\cup (M\cup R)&\;=\;\;&&(L\cup M)\cup (L\cup R)&&{\text{ (Left-distributivity of }}\,\cup \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\cap (M\cup R)&\;=\;\;&&(L\cap M)\cup (L\cap R)&&{\text{ (Left-distributivity of }}\,\cap \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\cap (M\cap R)&\;=\;\;&&(L\cap M)\cap (L\cap R)&&{\text{ (Left-distributivity of }}\,\cap \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\cap (M\,\triangle \,R)&\;=\;\;&&(L\cap M)\,\triangle \,(L\cap R)&&{\text{ (Left-distributivity of }}\,\cap \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]L\times (M\cap R)&\;=\;\;&&(L\times M)\cap (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\times (M\cup R)&\;=\;\;&&(L\times M)\cup (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times (M\,\setminus R)&\;=\;\;&&(L\times M)\,\setminus (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex]L\times (M\,\triangle R)&\;=\;\;&&(L\times M)\,\triangle (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]\end{alignedat}}}
Intersection distributes over symmetric difference:L∩(M△R)=(L∩M)△(L∩R){\displaystyle {\begin{alignedat}{5}L\,\cap \,(M\,\triangle \,R)~&~~=~~&&(L\,\cap \,M)\,\triangle \,(L\,\cap \,R)~&&~\\[1.4ex]\end{alignedat}}}(L△M)∩R=(L∩R)△(M∩R){\displaystyle {\begin{alignedat}{5}(L\,\triangle \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,\triangle \,(M\,\cap \,R)~&&~\\[1.4ex]\end{alignedat}}}
Union does not distribute over symmetric difference because only the following is guaranteed in general:L∪(M△R)⊇(L∪M)△(L∪R)=(M△R)∖L=(M∖L)△(R∖L){\displaystyle {\begin{alignedat}{5}L\cup (M\,\triangle \,R)~~{\color {red}{\supseteq }}~~\color {black}{\,}(L\cup M)\,\triangle \,(L\cup R)~&~=~&&(M\,\triangle \,R)\,\setminus \,L&~=~&&(M\,\setminus \,L)\,\triangle \,(R\,\setminus \,L)\\[1.4ex]\end{alignedat}}}
Symmetric difference does not distribute over itself:L△(M△R)≠(L△M)△(L△R)=M△R{\displaystyle L\,\triangle \,(M\,\triangle \,R)~~{\color {red}{\neq }}~~\color {black}{\,}(L\,\triangle \,M)\,\triangle \,(L\,\triangle \,R)~=~M\,\triangle \,R}and in general, for any setsLandA{\displaystyle L{\text{ and }}A}(whereA{\displaystyle A}representsM△R{\displaystyle M\,\triangle \,R}),L△A{\displaystyle L\,\triangle \,A}might not be a subset, nor a superset, ofL{\displaystyle L}(and the same is true forA{\displaystyle A}).
Failure of set subtraction to left distribute:
Set subtraction isrightdistributive over itself. However, set subtraction isnotleft distributive over itself because only the following is guaranteed in general:L∖(M∖R)⊇(L∖M)∖(L∖R)=L∩R∖M{\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\setminus \,R)&~~{\color {red}{\supseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\setminus \,(L\,\setminus \,R)~~=~~L\cap R\,\setminus \,M\\[1.4ex]\end{alignedat}}}where equality holds if and only ifL∖M=L∩R,{\displaystyle L\,\setminus \,M=L\,\cap \,R,}which happens if and only ifL∩M∩R=∅andL∖M⊆R.{\displaystyle L\cap M\cap R=\varnothing {\text{ and }}L\setminus M\subseteq R.}
For symmetric difference, the setsL∖(M△R){\displaystyle L\,\setminus \,(M\,\triangle \,R)}and(L∖M)△(L∖R)=L∩(M△R){\displaystyle (L\,\setminus \,M)\,\triangle \,(L\,\setminus \,R)=L\,\cap \,(M\,\triangle \,R)}are always disjoint.
So these two sets are equal if and only if they are both equal to∅.{\displaystyle \varnothing .}Moreover,L∖(M△R)=∅{\displaystyle L\,\setminus \,(M\,\triangle \,R)=\varnothing }if and only ifL∩M∩R=∅andL⊆M∪R.{\displaystyle L\cap M\cap R=\varnothing {\text{ and }}L\subseteq M\cup R.}
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related:(L∖M)∩(L∖R)=L∖(M∪R)⊆L∖(M∩R)=(L∖M)∪(L∖R){\displaystyle {\begin{alignedat}{5}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}L\,\setminus \,(M\,\cap \,R)~~=~~(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)\\[1.4ex]\end{alignedat}}}always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment⊆{\displaystyle {\color {red}{\subseteq }}}might be strict).
Equality holds if and only ifL∖(M∩R)⊆L∖(M∪R),{\displaystyle L\,\setminus \,(M\,\cap \,R)\;\subseteq \;L\,\setminus \,(M\,\cup \,R),}which happens if and only ifL∩M=L∩R.{\displaystyle L\,\cap \,M=L\,\cap \,R.}
This observation about De Morgan's laws shows that∖{\displaystyle \,\setminus \,}isnotleft distributive over∪{\displaystyle \,\cup \,}or∩{\displaystyle \,\cap \,}because only the following are guaranteed in general:L∖(M∪R)⊆(L∖M)∪(L∖R)=L∖(M∩R){\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cap \,R)\\[1.4ex]\end{alignedat}}}L∖(M∩R)⊇(L∖M)∩(L∖R)=L∖(M∪R){\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cap \,R)~&~~{\color {red}{\supseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)\\[1.4ex]\end{alignedat}}}where equality holds for one (or equivalently, for both) of the above two inclusion formulas if and only ifL∩M=L∩R.{\displaystyle L\,\cap \,M=L\,\cap \,R.}
The following statements are equivalent:
Quasi-commutativity:(L∖M)∖R=(L∖R)∖M(Quasi-commutative){\displaystyle (L\setminus M)\setminus R~=~(L\setminus R)\setminus M\qquad {\text{ (Quasi-commutative)}}}always holds but in general,L∖(M∖R)≠L∖(R∖M).{\displaystyle L\setminus (M\setminus R)~~{\color {red}{\neq }}~~L\setminus (R\setminus M).}However,L∖(M∖R)⊆L∖(R∖M){\displaystyle L\setminus (M\setminus R)~\subseteq ~L\setminus (R\setminus M)}if and only ifL∩R⊆M{\displaystyle L\cap R~\subseteq ~M}if and only ifL∖(R∖M)=L.{\displaystyle L\setminus (R\setminus M)~=~L.}
Set subtraction complexity: To manage the many identities involving set subtraction, this section is divided based on where the set subtraction operation and parentheses are located on the left hand side of the identity. The great variety and (relative) complexity of formulas involving set subtraction (compared to those without it) is in part due to the fact that unlike∪,∩,{\displaystyle \,\cup ,\,\cap ,}and△,{\displaystyle \triangle ,\,}set subtraction is neither associative nor commutative and it also is not left distributive over∪,∩,△,{\displaystyle \,\cup ,\,\cap ,\,\triangle ,}or even over itself.
Set subtraction isnotassociative in general:(L∖M)∖R≠L∖(M∖R){\displaystyle (L\,\setminus \,M)\,\setminus \,R\;~~{\color {red}{\neq }}~~\;L\,\setminus \,(M\,\setminus \,R)}since only the following is always guaranteed:(L∖M)∖R⊆L∖(M∖R).{\displaystyle (L\,\setminus \,M)\,\setminus \,R\;~~{\color {red}{\subseteq }}~~\;L\,\setminus \,(M\,\setminus \,R).}
(L∖M)∖R=L∖(M∪R)=(L∖R)∖M=(L∖M)∩(L∖R)=(L∖R)∖M=(L∖R)∖(M∖R){\displaystyle {\begin{alignedat}{4}(L\setminus M)\setminus R&=&&L\setminus (M\cup R)\\[0.6ex]&=(&&L\setminus R)\setminus M\\[0.6ex]&=(&&L\setminus M)\cap (L\setminus R)\\[0.6ex]&=(&&L\setminus R)\setminus M\\[0.6ex]&=(&&L\,\setminus \,R)\,\setminus \,(M\,\setminus \,R)\\[1.4ex]\end{alignedat}}}
L∖(M∖R)=(L∖M)∪(L∩R){\displaystyle {\begin{alignedat}{4}L\setminus (M\setminus R)&=(L\setminus M)\cup (L\cap R)\\[1.4ex]\end{alignedat}}}
Set subtraction on theleft, and parentheses on theleft
(L∖M)∪R=(L∪R)∖(M∖R)=(L∖(M∪R))∪R(the outermost union is disjoint){\displaystyle {\begin{alignedat}{4}\left(L\setminus M\right)\cup R&=(L\cup R)\setminus (M\setminus R)\\&=(L\setminus (M\cup R))\cup R~~~~~{\text{ (the outermost union is disjoint) }}\\\end{alignedat}}}
(L∖M)∩(L∖R)=L∖(M∪R)⊆L∖(M∩R)=(L∖M)∪(L∖R){\displaystyle {\begin{alignedat}{5}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}L\,\setminus \,(M\,\cap \,R)~~=~~(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)\\[1.4ex]\end{alignedat}}}(L∖M)△R=(L∖(M∪R))∪(R∖L)∪(L∩M∩R)(the three outermost sets are pairwise disjoint){\displaystyle {\begin{alignedat}{4}(L\setminus M)~\triangle ~R&=(L\setminus (M\cup R))\cup (R\setminus L)\cup (L\cap M\cap R)~~~{\text{ (the three outermost sets are pairwise disjoint) }}\\\end{alignedat}}}
(L∖M)×R=(L×R)∖(M×R)(Distributivity){\displaystyle (L\,\setminus M)\times R=(L\times R)\,\setminus (M\times R)~~~~~{\text{ (Distributivity)}}}
Set subtraction on theleft, and parentheses on theright
L∖(M∪R)=(L∖M)∩(L∖R)(De Morgan's law)=(L∖M)∖R=(L∖R)∖M{\displaystyle {\begin{alignedat}{3}L\setminus (M\cup R)&=(L\setminus M)&&\,\cap \,(&&L\setminus R)~~~~{\text{ (De Morgan's law) }}\\&=(L\setminus M)&&\,\,\setminus &&R\\&=(L\setminus R)&&\,\,\setminus &&M\\\end{alignedat}}}
L∖(M∩R)=(L∖M)∪(L∖R)(De Morgan's law){\displaystyle {\begin{alignedat}{4}L\setminus (M\cap R)&=(L\setminus M)\cup (L\setminus R)~~~~{\text{ (De Morgan's law) }}\\\end{alignedat}}}where the above two sets that are the subjects ofDe Morgan's lawsalways satisfyL∖(M∪R)⊆L∖(M∩R).{\displaystyle L\,\setminus \,(M\,\cup \,R)~~{\color {red}{\subseteq }}~~\color {black}{\,}L\,\setminus \,(M\,\cap \,R).}
L∖(M△R)=(L∖(M∪R))∪(L∩M∩R)(the outermost union is disjoint){\displaystyle {\begin{alignedat}{4}L\setminus (M~\triangle ~R)&=(L\setminus (M\cup R))\cup (L\cap M\cap R)~~~{\text{ (the outermost union is disjoint) }}\\\end{alignedat}}}
Set subtraction on theright, and parentheses on theleft
(L∪M)∖R=(L∖R)∪(M∖R){\displaystyle {\begin{alignedat}{4}(L\cup M)\setminus R&=(L\setminus R)\cup (M\setminus R)\\\end{alignedat}}}
(L∩M)∖R=(L∖R)∩(M∖R)=L∩(M∖R)=M∩(L∖R){\displaystyle {\begin{alignedat}{4}(L\cap M)\setminus R&=(&&L\setminus R)&&\cap (M\setminus R)\\&=&&L&&\cap (M\setminus R)\\&=&&M&&\cap (L\setminus R)\\\end{alignedat}}}
(L△M)∖R=(L∖R)△(M∖R)=(L∪R)△(M∪R){\displaystyle {\begin{alignedat}{4}(L\,\triangle \,M)\setminus R&=(L\setminus R)~&&\triangle ~(M\setminus R)\\&=(L\cup R)~&&\triangle ~(M\cup R)\\\end{alignedat}}}
Set subtraction on theright, and parentheses on theright
L∪(M∖R)=L∪(M∖(R∪L))(the outermost union is disjoint)=[(L∖M)∪(R∩L)]∪(M∖R)(the outermost union is disjoint)=(L∖(M∪R))∪(R∩L)∪(M∖R)(the three outermost sets are pairwise disjoint){\displaystyle {\begin{alignedat}{3}L\cup (M\setminus R)&=&&&&L&&\cup \;&&(M\setminus (R\cup L))&&~~~{\text{ (the outermost union is disjoint) }}\\&=[&&(&&L\setminus M)&&\cup \;&&(R\cap L)]\cup (M\setminus R)&&~~~{\text{ (the outermost union is disjoint) }}\\&=&&(&&L\setminus (M\cup R))\;&&\;\cup &&(R\cap L)\,\,\cup (M\setminus R)&&~~~{\text{ (the three outermost sets are pairwise disjoint) }}\\\end{alignedat}}}
L×(M∖R)=(L×M)∖(L×R)(Distributivity){\displaystyle L\times (M\,\setminus R)=(L\times M)\,\setminus (L\times R)~~~~~{\text{ (Distributivity)}}}
Operations of the form(L∙M)∗(M∙R){\displaystyle (L\bullet M)\ast (M\bullet R)}:
(L∪M)∪(M∪R)=L∪M∪R(L∪M)∩(M∪R)=M∪(L∩R)(L∪M)∖(M∪R)=L∖(M∪R)(L∪M)△(M∪R)=(L∖(M∪R))∪(R∖(L∪M))=(L△R)∖M(L∩M)∪(M∩R)=M∩(L∪R)(L∩M)∩(M∩R)=L∩M∩R(L∩M)∖(M∩R)=(L∩M)∖R(L∩M)△(M∩R)=[(L∩M)∪(M∩R)]∖(L∩M∩R)(L∖M)∪(M∖R)=(L∪M)∖(M∩R)(L∖M)∩(M∖R)=∅(L∖M)∖(M∖R)=L∖M(L∖M)△(M∖R)=(L∖M)∪(M∖R)=(L∪M)∖(M∩R)(L△M)∪(M△R)=(L∪M∪R)∖(L∩M∩R)(L△M)∩(M△R)=((L∩R)∖M)∪(M∖(L∪R))(L△M)∖(M△R)=(L∖(M∪R))∪((M∩R)∖L)(L△M)△(M△R)=L△R{\displaystyle {\begin{alignedat}{9}(L\cup M)&\,\cup \,&&(&&M\cup R)&&&&\;=\;\;&&L\cup M\cup R\\[1.4ex](L\cup M)&\,\cap \,&&(&&M\cup R)&&&&\;=\;\;&&M\cup (L\cap R)\\[1.4ex](L\cup M)&\,\setminus \,&&(&&M\cup R)&&&&\;=\;\;&&L\,\setminus \,(M\cup R)\\[1.4ex](L\cup M)&\,\triangle \,&&(&&M\cup R)&&&&\;=\;\;&&(L\,\setminus \,(M\cup R))\,\cup \,(R\,\setminus \,(L\cup M))\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\,\triangle \,R)\,\setminus \,M\\[1.4ex](L\cap M)&\,\cup \,&&(&&M\cap R)&&&&\;=\;\;&&M\cap (L\cup R)\\[1.4ex](L\cap M)&\,\cap \,&&(&&M\cap R)&&&&\;=\;\;&&L\cap M\cap R\\[1.4ex](L\cap M)&\,\setminus \,&&(&&M\cap R)&&&&\;=\;\;&&(L\cap M)\,\setminus \,R\\[1.4ex](L\cap M)&\,\triangle \,&&(&&M\cap R)&&&&\;=\;\;&&[(L\,\cap M)\cup (M\,\cap R)]\,\setminus \,(L\,\cap M\,\cap R)\\[1.4ex](L\,\setminus M)&\,\cup \,&&(&&M\,\setminus R)&&&&\;=\;\;&&(L\,\cup M)\,\setminus (M\,\cap \,R)\\[1.4ex](L\,\setminus M)&\,\cap \,&&(&&M\,\setminus R)&&&&\;=\;\;&&\varnothing \\[1.4ex](L\,\setminus M)&\,\setminus \,&&(&&M\,\setminus R)&&&&\;=\;\;&&L\,\setminus M\\[1.4ex](L\,\setminus M)&\,\triangle \,&&(&&M\,\setminus R)&&&&\;=\;\;&&(L\,\setminus M)\cup (M\,\setminus R)\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\,\cup M)\setminus (M\,\cap R)\\[1.4ex](L\,\triangle \,M)&\,\cup \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&(L\,\cup \,M\,\cup \,R)\,\setminus \,(L\,\cap \,M\,\cap \,R)\\[1.4ex](L\,\triangle \,M)&\,\cap \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&((L\,\cap \,R)\,\setminus \,M)\,\cup \,(M\,\setminus \,(L\,\cup \,R))\\[1.4ex](L\,\triangle \,M)&\,\setminus \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&(L\,\setminus \,(M\,\cup \,R))\,\cup \,((M\,\cap \,R)\,\setminus \,L)\\[1.4ex](L\,\triangle \,M)&\,\triangle \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&L\,\triangle \,R\\[1.7ex]\end{alignedat}}}
Operations of the form(L∙M)∗(R∖M){\displaystyle (L\bullet M)\ast (R\,\setminus \,M)}:
(L∪M)∪(R∖M)=L∪M∪R(L∪M)∩(R∖M)=(L∩R)∖M(L∪M)∖(R∖M)=M∪(L∖R)(L∪M)△(R∖M)=M∪(L△R)(L∩M)∪(R∖M)=[L∩(M∪R)]∪[R∖(L∪M)](disjoint union)=(L∩M)△(R∖M)(L∩M)∩(R∖M)=∅(L∩M)∖(R∖M)=L∩M(L∩M)△(R∖M)=(L∩M)∪(R∖M)(disjoint union)(L∖M)∪(R∖M)=L∪R∖M(L∖M)∩(R∖M)=(L∩R)∖M(L∖M)∖(R∖M)=L∖(M∪R)(L∖M)△(R∖M)=(L△R)∖M(L△M)∪(R∖M)=(L∪M∪R)∖(L∩M)(L△M)∩(R∖M)=(L∩R)∖M(L△M)∖(R∖M)=[L∖(M∪R)]∪(M∖L)(disjoint union)=(L△M)∖(L∩R)(L△M)△(R∖M)=L△(M∪R){\displaystyle {\begin{alignedat}{9}(L\cup M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\cup M\cup R\\[1.4ex](L\cup M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap R)\,\setminus \,M\\[1.4ex](L\cup M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&M\cup (L\,\setminus \,R)\\[1.4ex](L\cup M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&M\cup (L\,\triangle \,R)\\[1.4ex](L\cap M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&[L\cap (M\cup R)]\cup [R\,\setminus \,(L\cup M)]\qquad {\text{ (disjoint union)}}\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\cap M)\,\triangle \,(R\,\setminus \,M)\\[1.4ex](L\cap M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&\varnothing \\[1.4ex](L\cap M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\cap M\\[1.4ex](L\cap M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap M)\cup (R\,\setminus \,M)\qquad {\text{ (disjoint union)}}\\[1.4ex](L\,\setminus \,M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\cup R\,\setminus \,M\\[1.4ex](L\,\setminus \,M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap R)\,\setminus \,M\\[1.4ex](L\,\setminus \,M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\,\setminus \,(M\cup R)\\[1.4ex](L\,\setminus \,M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\,\triangle \,R)\,\setminus \,M\\[1.4ex](L\,\triangle \,M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cup M\cup R)\,\setminus \,(L\cap M)\\[1.4ex](L\,\triangle \,M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap R)\,\setminus \,M\\[1.4ex](L\,\triangle \,M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&[L\,\setminus \,(M\cup R)]\cup (M\,\setminus \,L)\qquad {\text{ (disjoint union)}}\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\,\triangle \,M)\setminus (L\,\cap R)\\[1.4ex](L\,\triangle \,M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\,\triangle \,(M\cup R)\\[1.7ex]\end{alignedat}}}
Operations of the form(L∖M)∗(L∖R){\displaystyle (L\,\setminus \,M)\ast (L\,\setminus \,R)}:
(L∖M)∪(L∖R)=L∖(M∩R)(L∖M)∩(L∖R)=L∖(M∪R)(L∖M)∖(L∖R)=(L∩R)∖M(L∖M)△(L∖R)=L∩(M△R)=(L∩M)△(L∩R){\displaystyle {\begin{alignedat}{9}(L\,\setminus M)&\,\cup \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cap \,R)\\[1.4ex](L\,\setminus M)&\,\cap \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cup \,R)\\[1.4ex](L\,\setminus M)&\,\setminus \,&&(&&L\,\setminus R)&&\;=\;&&(L\,\cap \,R)\,\setminus \,M\\[1.4ex](L\,\setminus M)&\,\triangle \,&&(&&L\,\setminus R)&&\;=\;&&L\,\cap \,(M\,\triangle \,R)\\[1.4ex]&\,&&\,&&\,&&\;=\;&&(L\cap M)\,\triangle \,(L\cap R)\\[1.4ex]\end{alignedat}}}
Other properties:
L∩M=RandL∩R=Mif and only ifM=R⊆L.{\displaystyle L\cap M=R\;{\text{ and }}\;L\cap R=M\qquad {\text{ if and only if }}\qquad M=R\subseteq L.}
Given finitely many setsL1,…,Ln,{\displaystyle L_{1},\ldots ,L_{n},}something belongs to theirsymmetric differenceif and only if it belongs to an odd number of these sets. Explicitly, for anyx,{\displaystyle x,}x∈L1△⋯△Ln{\displaystyle x\in L_{1}\triangle \cdots \triangle L_{n}}if and only if the cardinality|{i:x∈Li}|{\displaystyle \left|\left\{i:x\in L_{i}\right\}\right|}is odd. (Recall that symmetric difference is associative so parentheses are not needed for the setL1△⋯△Ln{\displaystyle L_{1}\triangle \cdots \triangle L_{n}}).
Consequently, the symmetric difference of three sets satisfies:L△M△R=(L∩M∩R)∪{x:xbelongs to exactly one of the setsL,M,R}(the union is disjoint)=[L∩M∩R]∪[L∖(M∪R)]∪[M∖(L∪R)]∪[R∖(L∪M)](all 4 sets enclosed by [ ] are pairwise disjoint){\displaystyle {\begin{alignedat}{4}L\,\triangle \,M\,\triangle \,R&=(L\cap M\cap R)\cup \{x:x{\text{ belongs to exactly one of the sets }}L,M,R\}~~~~~~{\text{ (the union is disjoint) }}\\&=[L\cap M\cap R]\cup [L\setminus (M\cup R)]\cup [M\setminus (L\cup R)]\cup [R\setminus (L\cup M)]~~~~~~~~~{\text{ (all 4 sets enclosed by [ ] are pairwise disjoint) }}\\\end{alignedat}}}
The binaryCartesian product⨯distributes overunions, intersections, set subtraction, and symmetric difference:
(L∩M)×R=(L×R)∩(M×R)(Right-distributivity of×over∩)(L∪M)×R=(L×R)∪(M×R)(Right-distributivity of×over∪)(L∖M)×R=(L×R)∖(M×R)(Right-distributivity of×over∖)(L△M)×R=(L×R)△(M×R)(Right-distributivity of×over△){\displaystyle {\begin{alignedat}{9}(L\,\cap \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cap \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cup \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\setminus \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\setminus \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\triangle \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]\end{alignedat}}}
L×(M∩R)=(L×M)∩(L×R)(Left-distributivity of×over∩)L×(M∪R)=(L×M)∪(L×R)(Left-distributivity of×over∪)L×(M∖R)=(L×M)∖(L×R)(Left-distributivity of×over∖)L×(M△R)=(L×M)△(L×R)(Left-distributivity of×over△){\displaystyle {\begin{alignedat}{5}L\times (M\cap R)&\;=\;\;&&(L\times M)\cap (L\times R)\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\times (M\cup R)&\;=\;\;&&(L\times M)\cup (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times (M\setminus R)&\;=\;\;&&(L\times M)\setminus (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex]L\times (M\triangle R)&\;=\;\;&&(L\times M)\triangle (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]\end{alignedat}}}
But in general, ⨯ does not distribute over itself:L×(M×R)≠(L×M)×(L×R){\displaystyle L\times (M\times R)~\color {Red}{\neq }\color {Black}{}~(L\times M)\times (L\times R)}(L×M)×R≠(L×R)×(M×R).{\displaystyle (L\times M)\times R~\color {Red}{\neq }\color {Black}{}~(L\times R)\times (M\times R).}
(L×R)∩(L2×R2)=(L∩L2)×(R∩R2){\displaystyle (L\times R)\cap \left(L_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(R\cap R_{2}\right)}(L×M×R)∩(L2×M2×R2)=(L∩L2)×(M∩M2)×(R∩R2){\displaystyle (L\times M\times R)\cap \left(L_{2}\times M_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)}
(L×R)∪(L2×R2)=[(L∖L2)×R]∪[(L2∖L)×R2]∪[(L∩L2)×(R∪R2)]=[L×(R∖R2)]∪[L2×(R2∖R)]∪[(L∪L2)×(R∩R2)]{\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\cup ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\setminus L_{2}\right)\times R\right]~\cup ~\left[\left(L_{2}\setminus L\right)\times R_{2}\right]~\cup ~\left[\left(L\cap L_{2}\right)\times \left(R\cup R_{2}\right)\right]\\[0.5ex]~&=~\left[L\times \left(R\setminus R_{2}\right)\right]~\cup ~\left[L_{2}\times \left(R_{2}\setminus R\right)\right]~\cup ~\left[\left(L\cup L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\\end{alignedat}}}
(L×R)∖(L2×R2)=[(L∖L2)×R]∪[L×(R∖R2)]{\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\setminus ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]~\cup ~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}}and(L×M×R)∖(L2×M2×R2)=[(L∖L2)×M×R]∪[L×(M∖M2)×R]∪[L×M×(R∖R2)]{\displaystyle (L\times M\times R)~\setminus ~\left(L_{2}\times M_{2}\times R_{2}\right)~=~\left[\left(L\,\setminus \,L_{2}\right)\times M\times R\right]~\cup ~\left[L\times \left(M\,\setminus \,M_{2}\right)\times R\right]~\cup ~\left[L\times M\times \left(R\,\setminus \,R_{2}\right)\right]}
(L∖L2)×(R∖R2)=(L×R)∖[(L2×R)∪(L×R2)]{\displaystyle \left(L\,\setminus \,L_{2}\right)\times \left(R\,\setminus \,R_{2}\right)~=~\left(L\times R\right)\,\setminus \,\left[\left(L_{2}\times R\right)\cup \left(L\times R_{2}\right)\right]}
(L∖L2)×(M∖M2)×(R∖R2)=(L×M×R)∖[(L2×M×R)∪(L×M2×R)∪(L×M×R2)]{\displaystyle \left(L\,\setminus \,L_{2}\right)\times \left(M\,\setminus \,M_{2}\right)\times \left(R\,\setminus \,R_{2}\right)~=~\left(L\times M\times R\right)\,\setminus \,\left[\left(L_{2}\times M\times R\right)\cup \left(L\times M_{2}\times R\right)\cup \left(L\times M\times R_{2}\right)\right]}
L×(R△R2)=[L×(R∖R2)]∪[L×(R2∖R)]{\displaystyle L\times \left(R\,\triangle \,R_{2}\right)~=~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\,\cup \,\left[L\times \left(R_{2}\,\setminus \,R\right)\right]}(L△L2)×R=[(L∖L2)×R]∪[(L2∖L)×R]{\displaystyle \left(L\,\triangle \,L_{2}\right)\times R~=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]\,\cup \,\left[\left(L_{2}\,\setminus \,L\right)\times R\right]}
(L△L2)×(R△R2)=[(L∪L2)×(R∪R2)]∖[((L∩L2)×R)∪(L×(R∩R2))]=[(L∖L2)×(R2∖R)]∪[(L2∖L)×(R2∖R)]∪[(L∖L2)×(R∖R2)]∪[(L2∖L)∪(R∖R2)]{\displaystyle {\begin{alignedat}{4}\left(L\,\triangle \,L_{2}\right)\times \left(R\,\triangle \,R_{2}\right)~&=~&&&&\,\left[\left(L\cup L_{2}\right)\times \left(R\cup R_{2}\right)\right]\;\setminus \;\left[\left(\left(L\cap L_{2}\right)\times R\right)\;\cup \;\left(L\times \left(R\cap R_{2}\right)\right)\right]\\[0.7ex]&=~&&&&\,\left[\left(L\,\setminus \,L_{2}\right)\times \left(R_{2}\,\setminus \,R\right)\right]\,\cup \,\left[\left(L_{2}\,\setminus \,L\right)\times \left(R_{2}\,\setminus \,R\right)\right]\,\cup \,\left[\left(L\,\setminus \,L_{2}\right)\times \left(R\,\setminus \,R_{2}\right)\right]\,\cup \,\left[\left(L_{2}\,\setminus \,L\right)\cup \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}}
(L△L2)×(M△M2)×(R△R2)=[(L∪L2)×(M∪M2)×(R∪R2)]∖[((L∩L2)×M×R)∪(L×(M∩M2)×R)∪(L×M×(R∩R2))]{\displaystyle {\begin{alignedat}{4}\left(L\,\triangle \,L_{2}\right)\times \left(M\,\triangle \,M_{2}\right)\times \left(R\,\triangle \,R_{2}\right)~&=~\left[\left(L\cup L_{2}\right)\times \left(M\cup M_{2}\right)\times \left(R\cup R_{2}\right)\right]\;\setminus \;\left[\left(\left(L\cap L_{2}\right)\times M\times R\right)\;\cup \;\left(L\times \left(M\cap M_{2}\right)\times R\right)\;\cup \;\left(L\times M\times \left(R\cap R_{2}\right)\right)\right]\\\end{alignedat}}}
In general,(L△L2)×(R△R2){\displaystyle \left(L\,\triangle \,L_{2}\right)\times \left(R\,\triangle \,R_{2}\right)}need not be a subset nor a superset of(L×R)△(L2×R2).{\displaystyle \left(L\times R\right)\,\triangle \,\left(L_{2}\times R_{2}\right).}
(L×R)△(L2×R2)=(L×R)∪(L2×R2)∖[(L∩L2)×(R∩R2)]{\displaystyle {\begin{alignedat}{4}\left(L\times R\right)\,\triangle \,\left(L_{2}\times R_{2}\right)~&=~&&\left(L\times R\right)\cup \left(L_{2}\times R_{2}\right)\;\setminus \;\left[\left(L\cap L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\[0.7ex]\end{alignedat}}}
(L×M×R)△(L2×M2×R2)=(L×M×R)∪(L2×M2×R2)∖[(L∩L2)×(M∩M2)×(R∩R2)]{\displaystyle {\begin{alignedat}{4}\left(L\times M\times R\right)\,\triangle \,\left(L_{2}\times M_{2}\times R_{2}\right)~&=~&&\left(L\times M\times R\right)\cup \left(L_{2}\times M_{2}\times R_{2}\right)\;\setminus \;\left[\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)\right]\\[0.7ex]\end{alignedat}}}
Let(Li)i∈I,{\displaystyle \left(L_{i}\right)_{i\in I},}(Rj)j∈J,{\displaystyle \left(R_{j}\right)_{j\in J},}and(Si,j)(i,j)∈I×J{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}}be indexedfamilies of sets. Whenever the assumption is needed, then allindexing sets, such asI{\displaystyle I}andJ,{\displaystyle J,}are assumed to be non-empty.
Afamily of setsor (more briefly) afamilyrefers to a set whose elements are sets.
Anindexed familyof setsis a function from some set, called itsindexing set, into some family of sets.
An indexed family of sets will be denoted by(Li)i∈I,{\displaystyle \left(L_{i}\right)_{i\in I},}where this notation assigns the symbolI{\displaystyle I}for the indexing set and for every indexi∈I,{\displaystyle i\in I,}assigns the symbolLi{\displaystyle L_{i}}to the value of the function ati.{\displaystyle i.}The function itself may then be denoted by the symbolL∙,{\displaystyle L_{\bullet },}which is obtained from the notation(Li)i∈I{\displaystyle \left(L_{i}\right)_{i\in I}}by replacing the indexi{\displaystyle i}with a bullet symbol∙;{\displaystyle \bullet \,;}explicitly,L∙{\displaystyle L_{\bullet }}is the function:L∙:I→{Li:i∈I}i↦Li{\displaystyle {\begin{alignedat}{4}L_{\bullet }:\;&&I&&\;\to \;&\left\{L_{i}:i\in I\right\}\\[0.3ex]&&i&&\;\mapsto \;&L_{i}\\\end{alignedat}}}which may be summarized by writingL∙=(Li)i∈I.{\displaystyle L_{\bullet }=\left(L_{i}\right)_{i\in I}.}
Any given indexed family of setsL∙=(Li)i∈I{\displaystyle L_{\bullet }=\left(L_{i}\right)_{i\in I}}(which is afunction) can be canonically associated with its image/rangeImL∙=def{Li:i∈I}{\displaystyle \operatorname {Im} L_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{L_{i}:i\in I\right\}}(which is a family of sets).
Conversely, any given family of setsB{\displaystyle {\mathcal {B}}}may be associated with theB{\displaystyle {\mathcal {B}}}-indexed family of sets(B)B∈B,{\displaystyle (B)_{B\in {\mathcal {B}}},}which is technically theidentity mapB→B.{\displaystyle {\mathcal {B}}\to {\mathcal {B}}.}However, this isnota bijective correspondence because an indexed family of setsL∙=(Li)i∈I{\displaystyle L_{\bullet }=\left(L_{i}\right)_{i\in I}}isnotrequired to be injective (that is, there may exist distinct indicesi≠j{\displaystyle i\neq j}such asLi=Lj{\displaystyle L_{i}=L_{j}}), which in particular means that it is possible for distinct indexed families of sets (which are functions) to be associated with the same family of sets (by having the same image/range).
Arbitrary unions defined[3]
IfI=∅{\displaystyle I=\varnothing }then⋃i∈∅Li={x:there existsi∈∅such thatx∈Li}=∅,{\displaystyle \bigcup _{i\in \varnothing }L_{i}=\{x~:~{\text{ there exists }}i\in \varnothing {\text{ such that }}x\in L_{i}\}=\varnothing ,}which is somethings called thenullary union convention(despite being called a convention, this equality follows from the definition).
IfB{\displaystyle {\mathcal {B}}}is a family of sets then∪B{\displaystyle \cup {\mathcal {B}}}denotes the set:⋃B=def⋃B∈BB=def{x:there existsB∈Bsuch thatx∈B}.{\displaystyle \bigcup {\mathcal {B}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcup _{B\in {\mathcal {B}}}B~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{x~:~{\text{ there exists }}B\in {\mathcal {B}}{\text{ such that }}x\in B\}.}
Arbitrary intersections defined
IfI≠∅{\displaystyle I\neq \varnothing }then[3]
IfB≠∅{\displaystyle {\mathcal {B}}\neq \varnothing }is anon-emptyfamily of sets then∩B{\displaystyle \cap {\mathcal {B}}}denotes the set:⋂B=def⋂B∈BB=def{x:x∈Bfor everyB∈B}={x:for allB,ifB∈Bthenx∈B}.{\displaystyle \bigcap {\mathcal {B}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcap _{B\in B}B~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{x~:~x\in B{\text{ for every }}B\in {\mathcal {B}}\}~=~\{x~:~{\text{ for all }}B,{\text{ if }}B\in {\mathcal {B}}{\text{ then }}x\in B\}.}
Nullary intersections
IfI=∅{\displaystyle I=\varnothing }then⋂i∈∅Li={x:for alli,ifi∈∅thenx∈Li}{\displaystyle \bigcap _{i\in \varnothing }L_{i}=\{x~:~{\text{ for all }}i,{\text{ if }}i\in \varnothing {\text{ then }}x\in L_{i}\}}where every possible thingx{\displaystyle x}in the universevacuouslysatisfied the condition: "ifi∈∅{\displaystyle i\in \varnothing }thenx∈Li{\displaystyle x\in L_{i}}". Consequently,⋂i∈∅Li={x:true}{\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}=\{x:{\text{ true }}\}}consists ofeverythingin the universe.
So ifI=∅{\displaystyle I=\varnothing }and:
A consequence of this is the following assumption/definition:
Some authors adopt the so callednullary intersectionconvention, which is the convention that an empty intersection of sets is equal to some canonical set. In particular, if all sets are subsets of some setX{\displaystyle X}then some author may declare that the empty intersection of these sets be equal toX.{\displaystyle X.}However, the nullary intersection convention is not as commonly accepted as the nullary union convention and this article will not adopt it (this is due to the fact that unlike the empty union, the value of the empty intersection depends onX{\displaystyle X}so if there are multiple sets under consideration, which is commonly the case, then the value of the empty intersection risks becoming ambiguous).
Multiple index sets⋃j∈Ji∈I,Si,j=def⋃(i,j)∈I×JSi,j{\displaystyle \bigcup _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcup _{(i,j)\in I\times J}S_{i,j}}⋂j∈Ji∈I,Si,j=def⋂(i,j)∈I×JSi,j{\displaystyle \bigcap _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcap _{(i,j)\in I\times J}S_{i,j}}
and[4]
and[4]
Naively swapping⋃i∈I{\displaystyle \;{\textstyle \bigcup \limits _{i\in I}}\;}and⋂j∈J{\displaystyle \;{\textstyle \bigcap \limits _{j\in J}}\;}may produce a different set
The following inclusion always holds:
In general, equality need not hold and moreover, the right hand side depends on how for each fixedi∈I,{\displaystyle i\in I,}the sets(Si,j)j∈J{\displaystyle \left(S_{i,j}\right)_{j\in J}}are labelled; and analogously, the left hand side depends on how for each fixedj∈J,{\displaystyle j\in J,}the sets(Si,j)i∈I{\displaystyle \left(S_{i,j}\right)_{i\in I}}are labelled. An example demonstrating this is now given.
Equality inInclusion 1 ∪∩ is a subset of ∩∪can hold under certain circumstances, such as in7e, which is the special case where(Si,j)(i,j)∈I×J{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}}is(Li∖Rj)(i,j)∈I×J{\displaystyle \left(L_{i}\setminus R_{j}\right)_{(i,j)\in I\times J}}(that is,Si,j:=Li∖Rj{\displaystyle S_{i,j}\colon =L_{i}\setminus R_{j}}with the same indexing setsI{\displaystyle I}andJ{\displaystyle J}), or such as in7f, which is the special case where(Si,j)(i,j)∈I×J{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}}is(Li∖Rj)(j,i)∈J×I{\displaystyle \left(L_{i}\setminus R_{j}\right)_{(j,i)\in J\times I}}(that is,S^j,i:=Li∖Rj{\displaystyle {\hat {S}}_{j,i}\colon =L_{i}\setminus R_{j}}with the indexing setsI{\displaystyle I}andJ{\displaystyle J}swapped).
For a correct formula that extends the distributive laws, an approach other than just switching∪{\displaystyle \cup }and∩{\displaystyle \cap }is needed.
Suppose that for eachi∈I,{\displaystyle i\in I,}Ji{\displaystyle J_{i}}is a non-empty index set and for eachj∈Ji,{\displaystyle j\in J_{i},}letTi,j{\displaystyle T_{i,j}}be any set (for example, to apply this law to(Si,j)(i,j)∈I×J,{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J},}useJi:=J{\displaystyle J_{i}\colon =J}for alli∈I{\displaystyle i\in I}and useTi,j:=Si,j{\displaystyle T_{i,j}\colon =S_{i,j}}for alli∈I{\displaystyle i\in I}and allj∈Ji=J{\displaystyle j\in J_{i}=J}). Let∏J∙=def∏i∈IJi{\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\prod _{i\in I}J_{i}}denote theCartesian product, which can be interpreted as the set of all functionsf:I→⋃i∈IJi{\displaystyle f~:~I~\to ~{\textstyle \bigcup \limits _{i\in I}}J_{i}}such thatf(i)∈Ji{\displaystyle f(i)\in J_{i}}for everyi∈I.{\displaystyle i\in I.}Such a function may also be denoted using the tuple notation(fi)i∈I{\displaystyle \left(f_{i}\right)_{i\in I}}wherefi=deff(i){\displaystyle f_{i}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(i)}for everyi∈I{\displaystyle i\in I}and conversely, a tuple(fi)i∈I{\displaystyle \left(f_{i}\right)_{i\in I}}is just notation for the function with domainI{\displaystyle I}whose value ati∈I{\displaystyle i\in I}isfi;{\displaystyle f_{i};}both notations can be used to denote the elements of∏J∙.{\displaystyle {\textstyle \prod }J_{\bullet }.}Then
where∏J∙=def∏i∈IJi.{\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}.}
Example application: In the particular case where allJi{\displaystyle J_{i}}are equal (that is,Ji=Ji2{\displaystyle J_{i}=J_{i_{2}}}for alli,i2∈I,{\displaystyle i,i_{2}\in I,}which is the case with the family(Si,j)(i,j)∈I×J,{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J},}for example), then lettingJ{\displaystyle J}denote this common set, the Cartesian product will be∏J∙=def∏i∈IJi=∏i∈IJ=JI,{\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}={\textstyle \prod \limits _{i\in I}}J=J^{I},}which is theset of all functionsof the formf:I→J.{\displaystyle f~:~I~\to ~J.}The above set equalitiesEq. 5 ∩∪ to ∪∩andEq. 6 ∪∩ to ∩∪, respectively become:[3]⋂i∈I⋃j∈JSi,j=⋃f∈JI⋂i∈ISi,f(i){\displaystyle \bigcap _{i\in I}\;\bigcup _{j\in J}S_{i,j}=\bigcup _{f\in J^{I}}\;\bigcap _{i\in I}S_{i,f(i)}}⋃i∈I⋂j∈JSi,j=⋂f∈JI⋃i∈ISi,f(i){\displaystyle \bigcup _{i\in I}\;\bigcap _{j\in J}S_{i,j}=\bigcap _{f\in J^{I}}\;\bigcup _{i\in I}S_{i,f(i)}}
which when combined withInclusion 1 ∪∩ is a subset of ∩∪implies:⋃i∈I⋂j∈JSi,j=⋂f∈JI⋃i∈ISi,f(i)⊆⋃g∈IJ⋂j∈JSg(j),j=⋂j∈J⋃i∈ISi,j{\displaystyle \bigcup _{i\in I}\;\bigcap _{j\in J}S_{i,j}~=~\bigcap _{f\in J^{I}}\;\bigcup _{i\in I}S_{i,f(i)}~~\color {Red}{\subseteq }\color {Black}{}~~\bigcup _{g\in I^{J}}\;\bigcap _{j\in J}S_{g(j),j}~=~\bigcap _{j\in J}\;\bigcup _{i\in I}S_{i,j}}where
Example application: To apply the general formula to the case of(Ck)k∈K{\displaystyle \left(C_{k}\right)_{k\in K}}and(Dl)l∈L,{\displaystyle \left(D_{l}\right)_{l\in L},}useI:={1,2},{\displaystyle I\colon =\{1,2\},}J1:=K,{\displaystyle J_{1}\colon =K,}J2:=L,{\displaystyle J_{2}\colon =L,}and letT1,k:=Ck{\displaystyle T_{1,k}\colon =C_{k}}for allk∈J1{\displaystyle k\in J_{1}}and letT2,l:=Dl{\displaystyle T_{2,l}\colon =D_{l}}for alll∈J2.{\displaystyle l\in J_{2}.}Every mapf∈∏J∙=def∏i∈IJi=J1×J2=K×L{\displaystyle f\in {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}=J_{1}\times J_{2}=K\times L}can bebijectivelyidentified with the pair(f(1),f(2))∈K×L{\displaystyle \left(f(1),f(2)\right)\in K\times L}(the inverse sends(k,l)∈K×L{\displaystyle (k,l)\in K\times L}to the mapf(k,l)∈∏J∙{\displaystyle f_{(k,l)}\in {\textstyle \prod }J_{\bullet }}defined by1↦k{\displaystyle 1\mapsto k}and2↦l;{\displaystyle 2\mapsto l;}this is technically just a change of notation). Recall thatEq. 5 ∩∪ to ∪∩was⋂i∈I⋃j∈JiTi,j=⋃f∈∏J∙⋂i∈ITi,f(i).{\displaystyle ~\bigcap _{i\in I}\;\bigcup _{j\in J_{i}}T_{i,j}=\bigcup _{f\in {\textstyle \prod }J_{\bullet }}\;\bigcap _{i\in I}T_{i,f(i)}.~}Expanding and simplifying the left hand side gives⋂i∈I⋃j∈JiTi,j=(⋃j∈J1T1,j)∩(⋃j∈J2T2,j)=(⋃k∈KT1,k)∩(⋃l∈LT2,l)=(⋃k∈KCk)∩(⋃l∈LDl){\displaystyle \bigcap _{i\in I}\;\bigcup _{j\in J_{i}}T_{i,j}=\left(\bigcup _{j\in J_{1}}T_{1,j}\right)\cap \left(\;\bigcup _{j\in J_{2}}T_{2,j}\right)=\left(\bigcup _{k\in K}T_{1,k}\right)\cap \left(\;\bigcup _{l\in L}T_{2,l}\right)=\left(\bigcup _{k\in K}C_{k}\right)\cap \left(\;\bigcup _{l\in L}D_{l}\right)}and doing the same to the right hand side gives:⋃f∈∏J∙⋂i∈ITi,f(i)=⋃f∈∏J∙(T1,f(1)∩T2,f(2))=⋃f∈∏J∙(Cf(1)∩Df(2))=⋃(k,l)∈K×L(Ck∩Dl)=⋃l∈Lk∈K,(Ck∩Dl).{\displaystyle \bigcup _{f\in \prod J_{\bullet }}\;\bigcap _{i\in I}T_{i,f(i)}=\bigcup _{f\in \prod J_{\bullet }}\left(T_{1,f(1)}\cap T_{2,f(2)}\right)=\bigcup _{f\in \prod J_{\bullet }}\left(C_{f(1)}\cap D_{f(2)}\right)=\bigcup _{(k,l)\in K\times L}\left(C_{k}\cap D_{l}\right)=\bigcup _{\stackrel {k\in K,}{l\in L}}\left(C_{k}\cap D_{l}\right).}
Thus the general identityEq. 5 ∩∪ to ∪∩reduces down to the previously given set equalityEq. 3b:(⋃k∈KCk)∩⋃l∈LDl=⋃l∈Lk∈K,(Ck∩Dl).{\displaystyle \left(\bigcup _{k\in K}C_{k}\right)\cap \;\bigcup _{l\in L}D_{l}=\bigcup _{\stackrel {k\in K,}{l\in L}}\left(C_{k}\cap D_{l}\right).}
The next identities are known asDe Morgan's laws.[4]
The following four set equalities can be deduced from the equalities7a-7dabove.
In general, naively swapping∪{\displaystyle \;\cup \;}and∩{\displaystyle \;\cap \;}may produce a different set (seethis notefor more details).
The equalities⋃i∈I⋂j∈J(Li∖Rj)=⋂j∈J⋃i∈I(Li∖Rj)and⋃j∈J⋂i∈I(Li∖Rj)=⋂i∈I⋃j∈J(Li∖Rj){\displaystyle \bigcup _{i\in I}\;\bigcap _{j\in J}\left(L_{i}\setminus R_{j}\right)~=~\bigcap _{j\in J}\;\bigcup _{i\in I}\left(L_{i}\setminus R_{j}\right)\quad {\text{ and }}\quad \bigcup _{j\in J}\;\bigcap _{i\in I}\left(L_{i}\setminus R_{j}\right)~=~\bigcap _{i\in I}\;\bigcup _{j\in J}\left(L_{i}\setminus R_{j}\right)}found inEq. 7eandEq. 7fare thus unusual in that they state exactly that swapping∪{\displaystyle \;\cup \;}and∩{\displaystyle \;\cap \;}willnotchange the resulting set.
Commutativity:[3]
⋃j∈Ji∈I,Si,j=def⋃(i,j)∈I×JSi,j=⋃i∈I(⋃j∈JSi,j)=⋃j∈J(⋃i∈ISi,j){\displaystyle \bigcup _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcup _{(i,j)\in I\times J}S_{i,j}~=~\bigcup _{i\in I}\left(\bigcup _{j\in J}S_{i,j}\right)~=~\bigcup _{j\in J}\left(\bigcup _{i\in I}S_{i,j}\right)}
⋂j∈Ji∈I,Si,j=def⋂(i,j)∈I×JSi,j=⋂i∈I(⋂j∈JSi,j)=⋂j∈J(⋂i∈ISi,j){\displaystyle \bigcap _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcap _{(i,j)\in I\times J}S_{i,j}~=~\bigcap _{i\in I}\left(\bigcap _{j\in J}S_{i,j}\right)~=~\bigcap _{j\in J}\left(\bigcap _{i\in I}S_{i,j}\right)}
Unions of unions and intersections of intersections:[3]
(⋃i∈ILi)∪R=⋃i∈I(Li∪R){\displaystyle \left(\bigcup _{i\in I}L_{i}\right)\cup R~=~\bigcup _{i\in I}\left(L_{i}\cup R\right)}(⋂i∈ILi)∩R=⋂i∈I(Li∩R){\displaystyle \left(\bigcap _{i\in I}L_{i}\right)\cap R~=~\bigcap _{i\in I}\left(L_{i}\cap R\right)}and[3]
and ifI=J{\displaystyle I=J}then also:[note 2][3]
If(Si,j)(i,j)∈I×J{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}}is a family of sets then
In particular, if(Li)i∈I{\displaystyle \left(L_{i}\right)_{i\in I}}and(Ri)i∈I{\displaystyle \left(R_{i}\right)_{i\in I}}are two families indexed by the same set then(∏i∈ILi)∩∏i∈IRi=∏i∈I(Li∩Ri){\displaystyle \left(\prod _{i\in I}L_{i}\right)\cap \prod _{i\in I}R_{i}~=~\prod _{i\in I}\left(L_{i}\cap R_{i}\right)}So for instance,(L×R)∩(L2×R2)=(L∩L2)×(R∩R2){\displaystyle (L\times R)\cap \left(L_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(R\cap R_{2}\right)}(L×R)∩(L2×R2)∩(L3×R3)=(L∩L2∩L3)×(R∩R2∩R3){\displaystyle (L\times R)\cap \left(L_{2}\times R_{2}\right)\cap \left(L_{3}\times R_{3}\right)~=~\left(L\cap L_{2}\cap L_{3}\right)\times \left(R\cap R_{2}\cap R_{3}\right)}and(L×M×R)∩(L2×M2×R2)=(L∩L2)×(M∩M2)×(R∩R2){\displaystyle (L\times M\times R)\cap \left(L_{2}\times M_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)}
Intersections of products indexed by different sets
Let(Li)i∈I{\displaystyle \left(L_{i}\right)_{i\in I}}and(Rj)j∈J{\displaystyle \left(R_{j}\right)_{j\in J}}be two families indexed by different sets.
Technically,I≠J{\displaystyle I\neq J}implies(∏i∈ILi)∩∏j∈JRj=∅.{\displaystyle \left({\textstyle \prod \limits _{i\in I}}L_{i}\right)\cap {\textstyle \prod \limits _{j\in J}}R_{j}=\varnothing .}However, sometimes these products are somehow identified as the same set through somebijectionor one of these products is identified as a subset of the other via someinjective map, in which case (byabuse of notation) this intersection may be equal to some other (possibly non-empty) set.
The binaryCartesian product⨯distributes overarbitrary intersections (when the indexing set is not empty) and over arbitrary unions:
L×(⋃i∈IRi)=⋃i∈I(L×Ri)(Left-distributivity of×over∪)L×(⋂i∈IRi)=⋂i∈I(L×Ri)(Left-distributivity of×over⋂i∈IwhenI≠∅)(⋃i∈ILi)×R=⋃i∈I(Li×R)(Right-distributivity of×over∪)(⋂i∈ILi)×R=⋂i∈I(Li×R)(Right-distributivity of×over⋂i∈IwhenI≠∅){\displaystyle {\begin{alignedat}{5}L\times \left(\bigcup _{i\in I}R_{i}\right)&\;=\;\;&&\bigcup _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times \left(\bigcap _{i\in I}R_{i}\right)&\;=\;\;&&\bigcap _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\left(\bigcup _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcup _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]\left(\bigcap _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcap _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\end{alignedat}}}
Suppose that for eachi∈I,{\displaystyle i\in I,}Ji{\displaystyle J_{i}}is a non-empty index set and for eachj∈Ji,{\displaystyle j\in J_{i},}letTi,j{\displaystyle T_{i,j}}be any set (for example, to apply this law to(Si,j)(i,j)∈I×J,{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J},}useJi:=J{\displaystyle J_{i}\colon =J}for alli∈I{\displaystyle i\in I}and useTi,j:=Si,j{\displaystyle T_{i,j}\colon =S_{i,j}}for alli∈I{\displaystyle i\in I}and allj∈Ji=J{\displaystyle j\in J_{i}=J}). Let∏J∙=def∏i∈IJi{\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\prod _{i\in I}J_{i}}denote theCartesian product, which (asmentioned above) can be interpreted as the set of all functionsf:I→⋃i∈IJi{\displaystyle f~:~I~\to ~{\textstyle \bigcup \limits _{i\in I}}J_{i}}such thatf(i)∈Ji{\displaystyle f(i)\in J_{i}}for everyi∈I{\displaystyle i\in I}.
Then
where∏J∙=def∏i∈IJi.{\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}.}
For unions, only the following is guaranteed in general:⋃j∈J∏i∈ISi,j⊆∏i∈I⋃j∈JSi,jand⋃i∈I∏j∈JSi,j⊆∏j∈J⋃i∈ISi,j{\displaystyle \bigcup _{j\in J}\;\prod _{i\in I}S_{i,j}~~\color {Red}{\subseteq }\color {Black}{}~~\prod _{i\in I}\;\bigcup _{j\in J}S_{i,j}\qquad {\text{ and }}\qquad \bigcup _{i\in I}\;\prod _{j\in J}S_{i,j}~~\color {Red}{\subseteq }\color {Black}{}~~\prod _{j\in J}\;\bigcup _{i\in I}S_{i,j}}where(Si,j)(i,j)∈I×J{\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}}is a family of sets.
However,(L×R)∪(L2×R2)=[(L∖L2)×R]∪[(L2∖L)×R2]∪[(L∩L2)×(R∪R2)]=[L×(R∖R2)]∪[L2×(R2∖R)]∪[(L∪L2)×(R∩R2)]{\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\cup ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\setminus L_{2}\right)\times R\right]~\cup ~\left[\left(L_{2}\setminus L\right)\times R_{2}\right]~\cup ~\left[\left(L\cap L_{2}\right)\times \left(R\cup R_{2}\right)\right]\\[0.5ex]~&=~\left[L\times \left(R\setminus R_{2}\right)\right]~\cup ~\left[L_{2}\times \left(R_{2}\setminus R\right)\right]~\cup ~\left[\left(L\cup L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\\end{alignedat}}}
If(Li)i∈I{\displaystyle \left(L_{i}\right)_{i\in I}}and(Ri)i∈I{\displaystyle \left(R_{i}\right)_{i\in I}}are two families of sets then:(∏i∈ILi)∖∏i∈IRi=⋃j∈I∏i∈I{Lj∖Rjifi=jLiifi≠j=⋃j∈I[(Lj∖Rj)×∏j≠ii∈I,Li]=⋃Lj⊈Rjj∈I,[(Lj∖Rj)×∏j≠ii∈I,Li]{\displaystyle {\begin{alignedat}{9}\left(\prod _{i\in I}L_{i}\right)~\setminus ~\prod _{i\in I}R_{i}~&=~\;~\bigcup _{j\in I}\;~\prod _{i\in I}{\begin{cases}L_{j}\,\setminus \,R_{j}&{\text{ if }}i=j\\L_{i}&{\text{ if }}i\neq j\\\end{cases}}\\[0.5ex]~&=~\;~\bigcup _{j\in I}\;~{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.5ex]~&=~\bigcup _{\stackrel {j\in I,}{L_{j}\not \subseteq R_{j}}}{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.3ex]\end{alignedat}}}so for instance,(L×R)∖(L2×R2)=[(L∖L2)×R]∪[L×(R∖R2)]{\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\setminus ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]~\cup ~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}}and(L×M×R)∖(L2×M2×R2)=[(L∖L2)×M×R]∪[L×(M∖M2)×R]∪[L×M×(R∖R2)]{\displaystyle (L\times M\times R)~\setminus ~\left(L_{2}\times M_{2}\times R_{2}\right)~=~\left[\left(L\,\setminus \,L_{2}\right)\times M\times R\right]~\cup ~\left[L\times \left(M\,\setminus \,M_{2}\right)\times R\right]~\cup ~\left[L\times M\times \left(R\,\setminus \,R_{2}\right)\right]}
(∏i∈ILi)△(∏i∈IRi)=(∏i∈ILi)∪(∏i∈IRi)∖∏i∈ILi∩Ri{\displaystyle {\begin{alignedat}{9}\left(\prod _{i\in I}L_{i}\right)~\triangle ~\left(\prod _{i\in I}R_{i}\right)~&=~\;~\left(\prod _{i\in I}L_{i}\right)~\cup ~\left(\prod _{i\in I}R_{i}\right)\;\setminus \;\prod _{i\in I}L_{i}\cap R_{i}\\[0.5ex]\end{alignedat}}}
Letf:X→Y{\displaystyle f:X\to Y}be any function.
LetLandR{\displaystyle L{\text{ and }}R}be completely arbitrary sets. AssumeA⊆XandC⊆Y.{\displaystyle A\subseteq X{\text{ and }}C\subseteq Y.}
Letf:X→Y{\displaystyle f:X\to Y}be any function, where we denote itsdomainX{\displaystyle X}bydomainf{\displaystyle \operatorname {domain} f}and denote itscodomainY{\displaystyle Y}bycodomainf.{\displaystyle \operatorname {codomain} f.}
Many of the identities below do not actually require that the sets be somehow related tof{\displaystyle f}'s domain or codomain (that is, toX{\displaystyle X}orY{\displaystyle Y}) so when some kind of relationship is necessary then it will be clearly indicated.
Because of this, in this article, ifL{\displaystyle L}is declared to be "any set," and it is not indicated thatL{\displaystyle L}must be somehow related toX{\displaystyle X}orY{\displaystyle Y}(say for instance, that it be a subsetX{\displaystyle X}orY{\displaystyle Y}) then it is meant thatL{\displaystyle L}is truly arbitrary.[note 3]This generality is useful in situations wheref:X→Y{\displaystyle f:X\to Y}is a map between two subsetsX⊆U{\displaystyle X\subseteq U}andY⊆V{\displaystyle Y\subseteq V}of some larger setsU{\displaystyle U}andV,{\displaystyle V,}and where the setL{\displaystyle L}might not be entirely contained inX=domainf{\displaystyle X=\operatorname {domain} f}and/orY=codomainf{\displaystyle Y=\operatorname {codomain} f}(e.g. if all that is known aboutL{\displaystyle L}is thatL⊆U{\displaystyle L\subseteq U}); in such a situation it may be useful to know what can and cannot be said aboutf(L){\displaystyle f(L)}and/orf−1(L){\displaystyle f^{-1}(L)}without having to introduce a (potentially unnecessary) intersection such as:f(L∩X){\displaystyle f(L\cap X)}and/orf−1(L∩Y).{\displaystyle f^{-1}(L\cap Y).}
Images and preimages of sets
IfL{\displaystyle L}isanyset then theimageofL{\displaystyle L}underf{\displaystyle f}is defined to be the set:f(L)=def{f(l):l∈L∩domainf}{\displaystyle f(L)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{\,f(l)~:~l\in L\cap \operatorname {domain} f\,\}}while thepreimageofL{\displaystyle L}underf{\displaystyle f}is:f−1(L)=def{x∈domainf:f(x)∈L}{\displaystyle f^{-1}(L)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{\,x\in \operatorname {domain} f~:~f(x)\in L\,\}}where ifL={s}{\displaystyle L=\{s\}}is a singleton set then thefiberorpreimageofs{\displaystyle s}underf{\displaystyle f}isf−1(s)=deff−1({s})={x∈domainf:f(x)=s}.{\displaystyle f^{-1}(s)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f^{-1}(\{s\})~=~\{\,x\in \operatorname {domain} f~:~f(x)=s\,\}.}
Denote byImf{\displaystyle \operatorname {Im} f}orimagef{\displaystyle \operatorname {image} f}theimageorrangeoff:X→Y,{\displaystyle f:X\to Y,}which is the set:Imf=deff(X)=deff(domainf)={f(x):x∈domainf}.{\displaystyle \operatorname {Im} f~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(X)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(\operatorname {domain} f)~=~\{f(x)~:~x\in \operatorname {domain} f\}.}
Saturated sets
A setA{\displaystyle A}is said to bef{\displaystyle f}-saturatedor asaturated setif any of the following equivalent conditions are satisfied:[3]
For a setA{\displaystyle A}to bef{\displaystyle f}-saturated, it is necessary thatA⊆domainf.{\displaystyle A\subseteq \operatorname {domain} f.}
Compositions and restrictions of functions
Iff{\displaystyle f}andg{\displaystyle g}are maps theng∘f{\displaystyle g\circ f}denotes thecompositionmapg∘f:{x∈domainf:f(x)∈domaing}→codomaing{\displaystyle g\circ f~:~\{\,x\in \operatorname {domain} f~:~f(x)\in \operatorname {domain} g\,\}~\to ~\operatorname {codomain} g}with domain and codomaindomain(g∘f)={x∈domainf:f(x)∈domaing}codomain(g∘f)=codomaing{\displaystyle {\begin{alignedat}{4}\operatorname {domain} (g\circ f)&=\{\,x\in \operatorname {domain} f~:~f(x)\in \operatorname {domain} g\,\}\\[0.4ex]\operatorname {codomain} (g\circ f)&=\operatorname {codomain} g\\[0.7ex]\end{alignedat}}}defined by(g∘f)(x)=defg(f(x)).{\displaystyle (g\circ f)(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~g(f(x)).}
Therestrictionoff:X→Y{\displaystyle f:X\to Y}toL,{\displaystyle L,}denoted byf|L,{\displaystyle f{\big \vert }_{L},}is the mapf|L:L∩domainf→Y{\displaystyle f{\big \vert }_{L}~:~L\cap \operatorname {domain} f~\to ~Y}withdomainf|L=L∩domainf{\displaystyle \operatorname {domain} f{\big \vert }_{L}~=~L\cap \operatorname {domain} f}defined by sendingx∈L∩domainf{\displaystyle x\in L\cap \operatorname {domain} f}tof(x);{\displaystyle f(x);}that is,f|L(x)=deff(x).{\displaystyle f{\big \vert }_{L}(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(x).}Alternatively,f|L=f∘In{\displaystyle ~f{\big \vert }_{L}~=~f\circ \operatorname {In} ~}whereIn:L∩X→X{\displaystyle ~\operatorname {In} ~:~L\cap X\to X~}denotes theinclusion map, which is defined byIn(s)=defs.{\displaystyle \operatorname {In} (s)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~s.}
If(Li)i∈I{\displaystyle \left(L_{i}\right)_{i\in I}}is a family of arbitrary sets indexed byI≠∅{\displaystyle I\neq \varnothing }then:[5]f(⋂i∈ILi)⊆⋂i∈If(Li)f(⋃i∈ILi)=⋃i∈If(Li)f−1(⋃i∈ILi)=⋃i∈If−1(Li)f−1(⋂i∈ILi)=⋂i∈If−1(Li){\displaystyle {\begin{alignedat}{4}f\left(\bigcap _{i\in I}L_{i}\right)\;&~\;\color {Red}{\subseteq }\color {Black}{}~\;\;\;\bigcap _{i\in I}f\left(L_{i}\right)\\f\left(\bigcup _{i\in I}L_{i}\right)\;&~=~\;\bigcup _{i\in I}f\left(L_{i}\right)\\f^{-1}\left(\bigcup _{i\in I}L_{i}\right)\;&~=~\;\bigcup _{i\in I}f^{-1}\left(L_{i}\right)\\f^{-1}\left(\bigcap _{i\in I}L_{i}\right)\;&~=~\;\bigcap _{i\in I}f^{-1}\left(L_{i}\right)\\\end{alignedat}}}
So of these four identities, it isonlyimages of intersectionsthat are not always preserved. Preimages preserve all basic set operations. Unions are preserved by both images and preimages.
If allLi{\displaystyle L_{i}}aref{\displaystyle f}-saturated then⋂i∈ILi{\displaystyle \bigcap _{i\in I}L_{i}}be will bef{\displaystyle f}-saturated and equality will hold in the first relation above; explicitly, this means:
If(Ai)i∈I{\displaystyle \left(A_{i}\right)_{i\in I}}is a family of arbitrary subsets ofX=domainf,{\displaystyle X=\operatorname {domain} f,}which means thatAi⊆X{\displaystyle A_{i}\subseteq X}for alli,{\displaystyle i,}thenConditional Equality 10abecomes:
Throughout, letL{\displaystyle L}andR{\displaystyle R}be any sets and letf:X→Y{\displaystyle f:X\to Y}be any function.
Summary
As the table below shows, set equality isnotguaranteedonlyforimagesof: intersections, set subtractions, and symmetric differences.
Preimages preserve set operations
Preimages of sets are well-behaved with respect to all basic set operations:
f−1(L∪R)=f−1(L)∪f−1(R)f−1(L∩R)=f−1(L)∩f−1(R)f−1(L∖R)=f−1(L)∖f−1(R)f−1(L△R)=f−1(L)△f−1(R){\displaystyle {\begin{alignedat}{4}f^{-1}(L\cup R)~&=~f^{-1}(L)\cup f^{-1}(R)\\f^{-1}(L\cap R)~&=~f^{-1}(L)\cap f^{-1}(R)\\f^{-1}(L\setminus \,R)~&=~f^{-1}(L)\setminus \,f^{-1}(R)\\f^{-1}(L\,\triangle \,R)~&=~f^{-1}(L)\,\triangle \,f^{-1}(R)\\\end{alignedat}}}
In words, preimagesdistribute overunions, intersections, set subtraction, and symmetric difference.
Imagesonlypreserve unions
Images of unions are well-behaved:
f(L∪R)=f(L)∪f(R){\displaystyle {\begin{alignedat}{4}f(L\cup R)~&=~f(L)\cup f(R)\\\end{alignedat}}}
but images of the other basic set operations arenotsince only the following are guaranteed in general:
f(L∩R)⊆f(L)∩f(R)f(L∖R)⊇f(L)∖f(R)f(L△R)⊇f(L)△f(R){\displaystyle {\begin{alignedat}{4}f(L\cap R)~&\subseteq ~f(L)\cap f(R)\\f(L\setminus R)~&\supseteq ~f(L)\setminus f(R)\\f(L\triangle R)~&\supseteq ~f(L)\,\triangle \,f(R)\\\end{alignedat}}}
In words, imagesdistribute overunions but not necessarily over intersections, set subtraction, or symmetric difference. What these latter three operations have in common is set subtraction: they eitherareset subtractionL∖R{\displaystyle L\setminus R}or else they can naturallybe definedas the set subtraction of two sets:L∩R=L∖(L∖R)andL△R=(L∪R)∖(L∩R).{\displaystyle L\cap R=L\setminus (L\setminus R)\quad {\text{ and }}\quad L\triangle R=(L\cup R)\setminus (L\cap R).}
IfL=X{\displaystyle L=X}thenf(X∖R)⊇f(X)∖f(R){\displaystyle f(X\setminus R)\supseteq f(X)\setminus f(R)}where as in the more general case, equality is not guaranteed. Iff{\displaystyle f}is surjective thenf(X∖R)⊇Y∖f(R),{\displaystyle f(X\setminus R)~\supseteq ~Y\setminus f(R),}which can be rewritten as:f(R∁)⊇f(R)∁{\displaystyle f\left(R^{\complement }\right)~\supseteq ~f(R)^{\complement }}ifR∁=defX∖R{\displaystyle R^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus R}andf(R)∁=defY∖f(R).{\displaystyle f(R)^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~Y\setminus f(R).}
Iff:{1,2}→Y{\displaystyle f:\{1,2\}\to Y}is constant,L={1},{\displaystyle L=\{1\},}andR={2}{\displaystyle R=\{2\}}then all four of the set containmentsf(L∩R)⊊f(L)∩f(R)f(L∖R)⊋f(L)∖f(R)f(X∖R)⊋f(X)∖f(R)f(L△R)⊋f(L)△f(R){\displaystyle {\begin{alignedat}{4}f(L\cap R)~&\subsetneq ~f(L)\cap f(R)\\f(L\setminus R)~&\supsetneq ~f(L)\setminus f(R)\\f(X\setminus R)~&\supsetneq ~f(X)\setminus f(R)\\f(L\triangle R)~&\supsetneq ~f(L)\triangle f(R)\\\end{alignedat}}}arestrict/proper(that is, the sets are not equal) since one side is the empty set while the other is non-empty. Thus equality is not guaranteed for even the simplest of functions.
The example above is now generalized to show that these four set equalities can fail for anyconstant functionwhose domain contains at least two (distinct) points.
Example:Letf:X→Y{\displaystyle f:X\to Y}be any constant function with imagef(X)={y}{\displaystyle f(X)=\{y\}}and suppose thatL,R⊆X{\displaystyle L,R\subseteq X}are non-empty disjoint subsets; that is,L≠∅,R≠∅,{\displaystyle L\neq \varnothing ,R\neq \varnothing ,}andL∩R=∅,{\displaystyle L\cap R=\varnothing ,}which implies that all of the setsL△R=L∪R,{\displaystyle L~\triangle ~R=L\cup R,}L∖R=L,{\displaystyle \,L\setminus R=L,}andX∖R⊇L∖R{\displaystyle X\setminus R\supseteq L\setminus R}are not empty and so consequently, their images underf{\displaystyle f}are all equal to{y}.{\displaystyle \{y\}.}
What the set operations in these four examples have in common is that they eitherareset subtraction∖{\displaystyle \setminus }(examples (1) and (2)) or else they can naturallybe definedas the set subtraction of two sets (examples (3) and (4)).
Mnemonic: In fact, for each of the above four set formulas for which equality is not guaranteed, the direction of the containment (that is, whether to use⊆or⊇{\displaystyle \,\subseteq {\text{ or }}\supseteq \,}) can always be deduced by imagining the functionf{\displaystyle f}as beingconstantand the two sets (L{\displaystyle L}andR{\displaystyle R}) as being non-empty disjoint subsets of its domain. This is becauseeveryequality fails for such a function and sets: one side will be always be∅{\displaystyle \varnothing }and the other non-empty − from this fact, the correct choice of⊆or⊇{\displaystyle \,\subseteq {\text{ or }}\supseteq \,}can be deduced by answering: "which side is empty?" For example, to decide if the?{\displaystyle ?}inf(L△R)∖f(R)?f((L△R)∖R){\displaystyle f(L\triangle R)\setminus f(R)~\;~?~\;~f((L\triangle R)\setminus R)}should be⊆or⊇,{\displaystyle \,\subseteq {\text{ or }}\supseteq ,\,}pretend[note 5]thatf{\displaystyle f}is constant and thatL△R{\displaystyle L\triangle R}andR{\displaystyle R}are non-empty disjoint subsets off{\displaystyle f}'s domain; then thelefthand side would be empty (sincef(L△R)∖f(R)={f's single value}∖{f's single value}=∅{\displaystyle f(L\triangle R)\setminus f(R)=\{f{\text{'s single value}}\}\setminus \{f{\text{'s single value}}\}=\varnothing }), which indicates that?{\displaystyle \,?\,}should be⊆{\displaystyle \,\subseteq \,}(the resulting statement is always guaranteed to be true) because this is the choice that will make∅=left hand side?right hand side{\displaystyle \varnothing ={\text{left hand side}}~\;~?~\;~{\text{right hand side}}}true.
Alternatively, the correct direction of containment can also be deduced by consideration of any constantf:{1,2}→Y{\displaystyle f:\{1,2\}\to Y}withL={1}{\displaystyle L=\{1\}}andR={2}.{\displaystyle R=\{2\}.}
Furthermore, this mnemonic can also be used to correctly deduce whether or not a set operation always distribute over images or preimages; for example, to determine whether or notf(L∩R){\displaystyle f(L\cap R)}always equalsf(L)∩f(R),{\displaystyle f(L)\cap f(R),}or alternatively, whether or notf−1(L∩R){\displaystyle f^{-1}(L\cap R)}always equalsf−1(L)∩f−1(R){\displaystyle f^{-1}(L)\cap f^{-1}(R)}(although∩{\displaystyle \,\cap \,}was used here, it can replaced by∪,∖,or△{\displaystyle \,\cup ,\,\setminus ,{\text{ or }}\,\triangle }). The answer to such a question can, as before, be deduced by consideration of this constant function: the answer for the general case (that is, for arbitraryf,L,{\displaystyle f,L,}andR{\displaystyle R}) is always the same as the answer for this choice of (constant) function and disjoint non-empty sets.
Characterizations of when equality holds forallsets:
For any functionf:X→Y,{\displaystyle f:X\to Y,}the following statements are equivalent:
In particular, if a map is not known to be injective then barring additional information, there is no guarantee that any of the equalities in statements (b) - (e) hold.
An example abovecan be used to help prove this characterization. Indeed, comparison of that example with such a proof suggests that the example is representative of the fundamental reason why one of these four equalities in statements (b) - (e) might not hold (that is, representative of "what goes wrong" when a set equality does not hold).
f(L∩R)⊆f(L)∩f(R)always holds{\displaystyle f(L\cap R)~\subseteq ~f(L)\cap f(R)\qquad \qquad {\text{ always holds}}}
Characterizations of equality: The following statements are equivalent:
Sufficient conditions for equality: Equality holds if any of the following are true:
In addition, the following always hold:f(f−1(L)∩R)=L∩f(R){\displaystyle f\left(f^{-1}(L)\cap R\right)~=~L\cap f(R)}f(f−1(L)∪R)=(L∩Imf)∪f(R){\displaystyle f\left(f^{-1}(L)\cup R\right)~=~(L\cap \operatorname {Im} f)\cup f(R)}
f(L∖R)⊇f(L)∖f(R)always holds{\displaystyle f(L\setminus R)~\supseteq ~f(L)\setminus f(R)\qquad \qquad {\text{ always holds}}}
Characterizations of equality: The following statements are equivalent:[proof 1]
Necessary conditions for equality(excluding characterizations): If equality holds then the following are necessarily true:
Sufficient conditions for equality: Equality holds if any of the following are true:
f(X∖R)⊇f(X)∖f(R)always holds, wheref:X→Y{\displaystyle f(X\setminus R)~\supseteq ~f(X)\setminus f(R)\qquad \qquad {\text{ always holds, where }}f:X\to Y}
Characterizations of equality: The following statements are equivalent:[proof 1]
where ifR⊆domainf{\displaystyle R\subseteq \operatorname {domain} f}then this list can be extended to include:
Sufficient conditions for equality: Equality holds if any of the following are true:
f(L△R)⊇f(L)△f(R)always holds{\displaystyle f\left(L~\triangle ~R\right)~\supseteq ~f(L)~\triangle ~f(R)\qquad \qquad {\text{ always holds}}}
Characterizations of equality: The following statements are equivalent:
Necessary conditions for equality(excluding characterizations): If equality holds then the following are necessarily true:
Sufficient conditions for equality: Equality holds if any of the following are true:
For any functionf:X→Y{\displaystyle f:X\to Y}and any setsL{\displaystyle L}andR,{\displaystyle R,}[proof 2]f(L∖R)=Y∖{y∈Y:L∩f−1(y)⊆R}=f(L)∖{y∈f(L):L∩f−1(y)⊆R}=f(L)∖{y∈f(L∩R):L∩f−1(y)⊆R}=f(L)∖{y∈V:L∩f−1(y)⊆R}for any supersetV⊇f(L∩R)=f(S)∖{y∈f(S):L∩f−1(y)⊆R}for any supersetS⊇L∩X.{\displaystyle {\begin{alignedat}{4}f(L\setminus R)&=Y~~~\;\,\,\setminus \left\{y\in Y~~~~~~~~~~\;\,~:~L\cap f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(L)\setminus \left\{y\in f(L)~~~~~~~\,~:~L\cap f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(L)\setminus \left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(L)\setminus \left\{y\in V~~~~~~~~~~~~\,~:~L\cap f^{-1}(y)\subseteq R\right\}\qquad &&{\text{ for any superset }}\quad V\supseteq f(L\cap R)\\[0.4ex]&=f(S)\setminus \left\{y\in f(S)~~~~~~~\,~:~L\cap f^{-1}(y)\subseteq R\right\}\qquad &&{\text{ for any superset }}\quad S\supseteq L\cap X.\\[0.7ex]\end{alignedat}}}
TakingL:=X=domainf{\displaystyle L:=X=\operatorname {domain} f}in the above formulas gives:f(X∖R)=Y∖{y∈Y:f−1(y)⊆R}=f(X)∖{y∈f(X):f−1(y)⊆R}=f(X)∖{y∈f(R):f−1(y)⊆R}=f(X)∖{y∈W:f−1(y)⊆R}for any supersetW⊇f(R){\displaystyle {\begin{alignedat}{4}f(X\setminus R)&=Y~~~\;\,\,\setminus \left\{y\in Y~~~~\;\,\,:~f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(X)\setminus \left\{y\in f(X)~:~f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(X)\setminus \left\{y\in f(R)~:~f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(X)\setminus \left\{y\in W~~~\;\,\,:~f^{-1}(y)\subseteq R\right\}\qquad {\text{ for any superset }}\quad W\supseteq f(R)\\[0.4ex]\end{alignedat}}}where the set{y∈f(R):f−1(y)⊆R}{\displaystyle \left\{y\in f(R):f^{-1}(y)\subseteq R\right\}}is equal to the image underf{\displaystyle f}of the largestf{\displaystyle f}-saturated subset ofR.{\displaystyle R.}
It follows fromL△R=(L∪R)∖(L∩R){\displaystyle L\,\triangle \,R=(L\cup R)\setminus (L\cap R)}and the above formulas for the image of a set subtraction that for any functionf:X→Y{\displaystyle f:X\to Y}and any setsL{\displaystyle L}andR,{\displaystyle R,}f(L△R)=Y∖{y∈Y:L∩f−1(y)=R∩f−1(y)}=f(L∪R)∖{y∈f(L∪R):L∩f−1(y)=R∩f−1(y)}=f(L∪R)∖{y∈f(L∩R):L∩f−1(y)=R∩f−1(y)}=f(L∪R)∖{y∈V:L∩f−1(y)=R∩f−1(y)}for any supersetV⊇f(L∩R)=f(S)∖{y∈f(S):L∩f−1(y)=R∩f−1(y)}for any supersetS⊇(L∪R)∩X.{\displaystyle {\begin{alignedat}{4}f(L\,\triangle \,R)&=Y~~~\;~~~\;~~~\;\setminus \left\{y\in Y~~~\,~~~\;~~~\,~~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\\[0.4ex]&=f(L\cup R)\setminus \left\{y\in f(L\cup R)~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\\[0.4ex]&=f(L\cup R)\setminus \left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\\[0.4ex]&=f(L\cup R)\setminus \left\{y\in V~~~\,~~~~~~~~~~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\qquad &&{\text{ for any superset }}\quad V\supseteq f(L\cap R)\\[0.4ex]&=f(S)~~\,~~~\,~\,\setminus \left\{y\in f(S)~~~\,~~~\;~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\qquad &&{\text{ for any superset }}\quad S\supseteq (L\cup R)\cap X.\\[0.7ex]\end{alignedat}}}
It follows from the above formulas for the image of a set subtraction that for any functionf:X→Y{\displaystyle f:X\to Y}and any setL,{\displaystyle L,}f(L)=Y∖{y∈Y:f−1(y)∩L=∅}=Imf∖{y∈Imf:f−1(y)∩L=∅}=W∖{y∈W:f−1(y)∩L=∅}for any supersetW⊇f(L){\displaystyle {\begin{alignedat}{4}f(L)&=Y~~~\;\,\setminus \left\{y\in Y~~~\;\,~:~f^{-1}(y)\cap L=\varnothing \right\}\\[0.4ex]&=\operatorname {Im} f\setminus \left\{y\in \operatorname {Im} f~:~f^{-1}(y)\cap L=\varnothing \right\}\\[0.4ex]&=W~~~\,\setminus \left\{y\in W~~\;\,~:~f^{-1}(y)\cap L=\varnothing \right\}\qquad {\text{ for any superset }}\quad W\supseteq f(L)\\[0.7ex]\end{alignedat}}}
This is more easily seen as being a consequence of the fact that for anyy∈Y,{\displaystyle y\in Y,}f−1(y)∩L=∅{\displaystyle f^{-1}(y)\cap L=\varnothing }if and only ify∉f(L).{\displaystyle y\not \in f(L).}
It follows from the above formulas for the image of a set that for any functionf:X→Y{\displaystyle f:X\to Y}and any setsL{\displaystyle L}andR,{\displaystyle R,}f(L∩R)=Y∖{y∈Y:L∩R∩f−1(y)=∅}=f(L)∖{y∈f(L):L∩R∩f−1(y)=∅}=f(L)∖{y∈U:L∩R∩f−1(y)=∅}for any supersetU⊇f(L)=f(R)∖{y∈f(R):L∩R∩f−1(y)=∅}=f(R)∖{y∈V:L∩R∩f−1(y)=∅}for any supersetV⊇f(R)=f(L)∩f(R)∖{y∈f(L)∩f(R):L∩R∩f−1(y)=∅}{\displaystyle {\begin{alignedat}{4}f(L\cap R)&=Y~~~~~\setminus \left\{y\in Y~~~~~~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.4ex]&=f(L)\setminus \left\{y\in f(L)~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.4ex]&=f(L)\setminus \left\{y\in U~~~~~~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}\qquad &&{\text{ for any superset }}\quad U\supseteq f(L)\\[0.4ex]&=f(R)\setminus \left\{y\in f(R)~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.4ex]&=f(R)\setminus \left\{y\in V~~~~~~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}\qquad &&{\text{ for any superset }}\quad V\supseteq f(R)\\[0.4ex]&=f(L)\cap f(R)\setminus \left\{y\in f(L)\cap f(R)~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.7ex]\end{alignedat}}}where moreover, for anyy∈Y,{\displaystyle y\in Y,}
The setsU{\displaystyle U}andV{\displaystyle V}mentioned above could, in particular, be any of the setsf(L∪R),Imf,{\displaystyle f(L\cup R),\;\operatorname {Im} f,}orY,{\displaystyle Y,}for example.
LetL{\displaystyle L}andR{\displaystyle R}be arbitrary sets,f:X→Y{\displaystyle f:X\to Y}be any map, and letA⊆X{\displaystyle A\subseteq X}andC⊆Y.{\displaystyle C\subseteq Y.}
Equality holds if any of the following are true:
(Pre)Images of operations on images
Sincef(L)∖f(L∖R)={y∈f(L∩R):L∩f−1(y)⊆R},{\displaystyle f(L)\setminus f(L\setminus R)~=~\left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)\subseteq R\right\},}
f−1(f(L)∖f(L∖R))=f−1({y∈f(L∩R):L∩f−1(y)⊆R})={x∈f−1(f(L∩R)):L∩f−1(f(x))⊆R}{\displaystyle {\begin{alignedat}{4}f^{-1}(f(L)\setminus f(L\setminus R))&=&&f^{-1}\left(\left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)\subseteq R\right\}\right)\\&=&&\left\{x\in f^{-1}(f(L\cap R))~:~L\cap f^{-1}(f(x))\subseteq R\right\}\\\end{alignedat}}}
Sincef(X)∖f(L∖R)={y∈f(X):L∩f−1(y)⊆R},{\displaystyle f(X)\setminus f(L\setminus R)~=~\left\{y\in f(X)~:~L\cap f^{-1}(y)\subseteq R\right\},}f−1(Y∖f(L∖R))=f−1(f(X)∖f(L∖R))=f−1({y∈f(X):L∩f−1(y)⊆R})={x∈X:L∩f−1(f(x))⊆R}=X∖f−1(f(L∖R)){\displaystyle {\begin{alignedat}{4}f^{-1}(Y\setminus f(L\setminus R))&~=~&&f^{-1}(f(X)\setminus f(L\setminus R))\\&=&&f^{-1}\left(\left\{y\in f(X)~:~L\cap f^{-1}(y)\subseteq R\right\}\right)\\&=&&\left\{x\in X~:~L\cap f^{-1}(f(x))\subseteq R\right\}\\&~=~&&X\setminus f^{-1}(f(L\setminus R))\\\end{alignedat}}}
UsingL:=X,{\displaystyle L:=X,}this becomesf(X)∖f(X∖R)={y∈f(R):f−1(y)⊆R}{\displaystyle ~f(X)\setminus f(X\setminus R)~=~\left\{y\in f(R)~:~f^{-1}(y)\subseteq R\right\}~}andf−1(Y∖f(X∖R))=f−1(f(X)∖f(X∖R))=f−1({y∈f(R):f−1(y)⊆R})={r∈R∩X:f−1(f(r))⊆R}⊆R{\displaystyle {\begin{alignedat}{4}f^{-1}(Y\setminus f(X\setminus R))&~=~&&f^{-1}(f(X)\setminus f(X\setminus R))\\&=&&f^{-1}\left(\left\{y\in f(R)~:~f^{-1}(y)\subseteq R\right\}\right)\\&=&&\left\{r\in R\cap X~:~f^{-1}(f(r))\subseteq R\right\}\\&\subseteq &&R\\\end{alignedat}}}and sof−1(Y∖f(L))=f−1(f(X)∖f(L))=f−1({y∈f(X∖L):f−1(y)∩L=∅})={x∈X∖L:f(x)∉f(L)}=X∖f−1(f(L))⊆X∖L{\displaystyle {\begin{alignedat}{4}f^{-1}(Y\setminus f(L))&~=~&&f^{-1}(f(X)\setminus f(L))\\&=&&f^{-1}\left(\left\{y\in f(X\setminus L)~:~f^{-1}(y)\cap L=\varnothing \right\}\right)\\&=&&\{x\in X\setminus L~:~f(x)\not \in f(L)\}\\&=&&X\setminus f^{-1}(f(L))\\&\subseteq &&X\setminus L\\\end{alignedat}}}
Let∏Y∙=def∏j∈JYj{\displaystyle \prod Y_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\prod _{j\in J}Y_{j}}and for everyk∈J,{\displaystyle k\in J,}letπk:∏j∈JYj→Yk{\displaystyle \pi _{k}~:~\prod _{j\in J}Y_{j}~\to ~Y_{k}}denote the canonical projection ontoYk.{\displaystyle Y_{k}.}
Definitions
Given a collection of mapsFj:X→Yj{\displaystyle F_{j}:X\to Y_{j}}indexed byj∈J,{\displaystyle j\in J,}define the map(Fj)j∈J:X→∏j∈JYjx↦(Fj(xj))j∈J,{\displaystyle {\begin{alignedat}{4}\left(F_{j}\right)_{j\in J}:\;&&X&&\;\to \;&\prod _{j\in J}Y_{j}\\[0.3ex]&&x&&\;\mapsto \;&\left(F_{j}\left(x_{j}\right)\right)_{j\in J},\\\end{alignedat}}}which is also denoted byF∙=(Fj)j∈J.{\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}.}This is the unique map satisfyingπj∘F∙=Fjfor allj∈J.{\displaystyle \pi _{j}\circ F_{\bullet }=F_{j}\quad {\text{ for all }}j\in J.}
Conversely, if given a mapF:X→∏j∈JYj{\displaystyle F~:~X~\to ~\prod _{j\in J}Y_{j}}thenF=(πj∘F)j∈J.{\displaystyle F=\left(\pi _{j}\circ F\right)_{j\in J}.}Explicitly, what this means is that ifFk=defπk∘F:X→Yk{\displaystyle F_{k}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\pi _{k}\circ F~:~X~\to ~Y_{k}}is defined for everyk∈J,{\displaystyle k\in J,}thenF{\displaystyle F}the unique map satisfying:πj∘F=Fj{\displaystyle \pi _{j}\circ F=F_{j}}for allj∈J;{\displaystyle j\in J;}or said more briefly,F=(Fj)j∈J.{\displaystyle F=\left(F_{j}\right)_{j\in J}.}
The mapF∙=(Fj)j∈J:X→∏j∈JYj{\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}}should not be confused with theCartesian product∏j∈JFj{\displaystyle \prod _{j\in J}F_{j}}of these maps, which is by definition is the map∏j∈JFj:∏j∈JX→∏j∈JYj(xj)j∈J↦(Fj(xj))j∈J{\displaystyle {\begin{alignedat}{4}\prod _{j\in J}F_{j}:\;&&\prod _{j\in J}X&&~\;\to \;~&\prod _{j\in J}Y_{j}\\[0.3ex]&&\left(x_{j}\right)_{j\in J}&&~\;\mapsto \;~&\left(F_{j}\left(x_{j}\right)\right)_{j\in J}\\\end{alignedat}}}with domain∏j∈JX=XJ{\displaystyle \prod _{j\in J}X=X^{J}}rather thanX.{\displaystyle X.}
Preimage and images of a Cartesian product
SupposeF∙=(Fj)j∈J:X→∏j∈JYj.{\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}.}
IfA⊆X{\displaystyle A~\subseteq ~X}thenF∙(A)⊆∏j∈JFj(A).{\displaystyle F_{\bullet }(A)~~\;\color {Red}{\subseteq }\color {Black}{}\;~~\prod _{j\in J}F_{j}(A).}
IfB⊆∏j∈JYj{\displaystyle B~\subseteq ~\prod _{j\in J}Y_{j}}thenF∙−1(B)⊆⋂j∈JFj−1(πj(B)){\displaystyle F_{\bullet }^{-1}(B)~~\;\color {Red}{\subseteq }\color {Black}{}\;~~\bigcap _{j\in J}F_{j}^{-1}\left(\pi _{j}(B)\right)}where equality will hold ifB=∏j∈Jπj(B),{\displaystyle B=\prod _{j\in J}\pi _{j}(B),}in which caseF∙−1(B)=⋂j∈JFj−1(πj(B)){\textstyle F_{\bullet }^{-1}(B)=\displaystyle \bigcap _{j\in J}F_{j}^{-1}\left(\pi _{j}(B)\right)}and
For equality to hold, it suffices for there to exist a family(Bj)j∈J{\displaystyle \left(B_{j}\right)_{j\in J}}of subsetsBj⊆Yj{\displaystyle B_{j}\subseteq Y_{j}}such thatB=∏j∈JBj,{\displaystyle B=\prod _{j\in J}B_{j},}in which case:
andπj(B)=Bj{\displaystyle \pi _{j}(B)=B_{j}}for allj∈J.{\displaystyle j\in J.}
Equivalences and implications of images and preimages
IfC⊆Imf{\displaystyle C~\subseteq ~\operatorname {Im} f}thenf−1(C)⊆f−1(R){\displaystyle f^{-1}(C)~\subseteq ~f^{-1}(R)}if and only ifC⊆R.{\displaystyle C~\subseteq ~R.}
The following are equivalent whenA⊆X:{\displaystyle A\subseteq X:}
Equality holds ifand only ifthe following is true:
Equality holds if any of the following are true:
Equality holds ifand only ifthe following is true:
Equality holds if any of the following are true:
Intersection of a set and a (pre)image
The following statements are equivalent:
Thus for anyt,{\displaystyle t,}[5]t∉f(L)if and only ifL∩f−1(t)=∅.{\displaystyle t\not \in f(L)\quad {\text{ if and only if }}\quad L\cap f^{-1}(t)=\varnothing .}
Afamily of setsor simply afamilyis a set whose elements are sets.
Afamily overX{\displaystyle X}is a family of subsets ofX.{\displaystyle X.}
Thepower setof a setX{\displaystyle X}is the set of all subsets ofX{\displaystyle X}:℘(X):={S:S⊆X}.{\displaystyle \wp (X)~\colon =~\{\;S~:~S\subseteq X\;\}.}
Notation for sequences of sets
Throughout,SandT{\displaystyle S{\text{ and }}T}will be arbitrary sets andS∙{\displaystyle S_{\bullet }}and will denote anetor asequenceof sets where if it is a sequence then this will be indicated by either of the notationsS∙=(Si)i=1∞orS∙=(Si)i∈N{\displaystyle S_{\bullet }=\left(S_{i}\right)_{i=1}^{\infty }\qquad {\text{ or }}\qquad S_{\bullet }=\left(S_{i}\right)_{i\in \mathbb {N} }}whereN{\displaystyle \mathbb {N} }denotes thenatural numbers.
A notationS∙=(Si)i∈I{\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}}indicates thatS∙{\displaystyle S_{\bullet }}is anetdirectedby(I,≤),{\displaystyle (I,\leq ),}which (by definition) is asequenceif the setI,{\displaystyle I,}which is called the net'sindexing set, is the natural numbers (that is, ifI=N{\displaystyle I=\mathbb {N} }) and≤{\displaystyle \,\leq \,}is the natural order onN.{\displaystyle \mathbb {N} .}
Disjoint and monotone sequences of sets
IfSi∩Sj=∅{\displaystyle S_{i}\cap S_{j}=\varnothing }for all distinct indicesi≠j{\displaystyle i\neq j}thenS∙{\displaystyle S_{\bullet }}is called apairwise disjointor simply adisjoint.
A sequence or netS∙{\displaystyle S_{\bullet }}of set is calledincreasingornon-decreasingif (resp.decreasingornon-increasing) if for all indicesi≤j,{\displaystyle i\leq j,}Si⊆Sj{\displaystyle S_{i}\subseteq S_{j}}(resp.Si⊇Sj{\displaystyle S_{i}\supseteq S_{j}}).
A sequence or netS∙{\displaystyle S_{\bullet }}of set is calledstrictly increasing(resp.strictly decreasing) if it is non-decreasing (resp. is non-increasing) and alsoSi≠Sj{\displaystyle S_{i}\neq S_{j}}for alldistinctindicesiandj.{\displaystyle i{\text{ and }}j.}It is calledmonotoneif it is non-decreasing or non-increasing and it is calledstrictly monotoneif it is strictly increasing or strictly decreasing.
A sequences or netS∙{\displaystyle S_{\bullet }}is said toincrease toS,{\displaystyle S,}denoted byS∙↑S{\displaystyle S_{\bullet }\uparrow S}[11]orS∙↗S,{\displaystyle S_{\bullet }\nearrow S,}ifS∙{\displaystyle S_{\bullet }}is increasing and the union of allSi{\displaystyle S_{i}}isS;{\displaystyle S;}that is, if⋃nSn=SandSi⊆Sjwheneveri≤j.{\displaystyle \bigcup _{n}S_{n}=S\qquad {\text{ and }}\qquad S_{i}\subseteq S_{j}\quad {\text{ whenever }}i\leq j.}It is said todecrease toS,{\displaystyle S,}denoted byS∙↓S{\displaystyle S_{\bullet }\downarrow S}[11]orS∙↘S,{\displaystyle S_{\bullet }\searrow S,}ifS∙{\displaystyle S_{\bullet }}is increasing and the intersection of allSi{\displaystyle S_{i}}isS{\displaystyle S}that is, if⋂nSn=SandSi⊇Sjwheneveri≤j.{\displaystyle \bigcap _{n}S_{n}=S\qquad {\text{ and }}\qquad S_{i}\supseteq S_{j}\quad {\text{ whenever }}i\leq j.}
Definitions of elementwise operations on families
IfLandR{\displaystyle {\mathcal {L}}{\text{ and }}{\mathcal {R}}}are families of sets and ifS{\displaystyle S}is any set then define:[12]L(∪)R:={L∪R:L∈LandR∈R}{\displaystyle {\mathcal {L}}\;(\cup )\;{\mathcal {R}}~\colon =~\{~L\cup R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}}L(∩)R:={L∩R:L∈LandR∈R}{\displaystyle {\mathcal {L}}\;(\cap )\;{\mathcal {R}}~\colon =~\{~L\cap R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}}L(∖)R:={L∖R:L∈LandR∈R}{\displaystyle {\mathcal {L}}\;(\setminus )\;{\mathcal {R}}~\colon =~\{~L\setminus R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}}L(△)R:={L△R:L∈LandR∈R}{\displaystyle {\mathcal {L}}\;(\triangle )\;{\mathcal {R}}~\colon =~\{~L\;\triangle \;R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}}L|S:={L∩S:L∈L}=L(∩){S}{\displaystyle {\mathcal {L}}{\big \vert }_{S}~\colon =~\{L\cap S~:~L\in {\mathcal {L}}\}={\mathcal {L}}\;(\cap )\;\{S\}}which are respectively calledelementwiseunion,elementwiseintersection,elementwise(set)difference,elementwisesymmetric difference, and thetrace/restriction ofL{\displaystyle {\mathcal {L}}}toS.{\displaystyle S.}The regular union, intersection, and set difference are all defined as usual and are denoted with their usual notation:L∪R,L∩R,L△R,{\displaystyle {\mathcal {L}}\cup {\mathcal {R}},{\mathcal {L}}\cap {\mathcal {R}},{\mathcal {L}}\;\triangle \;{\mathcal {R}},}andL∖R,{\displaystyle {\mathcal {L}}\setminus {\mathcal {R}},}respectively.
These elementwise operations on families of sets play an important role in, among other subjects, the theory offiltersand prefilters on sets.
Theupward closureinX{\displaystyle X}of a familyL⊆℘(X){\displaystyle {\mathcal {L}}\subseteq \wp (X)}is the family:L↑X:=⋃L∈L{S:L⊆S⊆X}={S⊆X:there existsL∈Lsuch thatL⊆S}{\displaystyle {\mathcal {L}}^{\uparrow X}~\colon =~\bigcup _{L\in {\mathcal {L}}}\{\;S~:~L\subseteq S\subseteq X\;\}~=~\{\;S\subseteq X~:~{\text{ there exists }}L\in {\mathcal {L}}{\text{ such that }}L\subseteq S\;\}}and thedownward closure ofL{\displaystyle {\mathcal {L}}}is the family:L↓:=⋃L∈L℘(L)={S:there existsL∈Lsuch thatS⊆L}.{\displaystyle {\mathcal {L}}^{\downarrow }~\colon =~\bigcup _{L\in {\mathcal {L}}}\wp (L)~=~\{\;S~:~{\text{ there exists }}L\in {\mathcal {L}}{\text{ such that }}S\subseteq L\;\}.}
The following table lists some well-known categories of families of sets having applications ingeneral topologyandmeasure theory.
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
A familyL{\displaystyle {\mathcal {L}}}is calledisotone,ascending, orupward closedinX{\displaystyle X}ifL⊆℘(X){\displaystyle {\mathcal {L}}\subseteq \wp (X)}andL=L↑X.{\displaystyle {\mathcal {L}}={\mathcal {L}}^{\uparrow X}.}[12]A familyL{\displaystyle {\mathcal {L}}}is calleddownward closedifL=L↓.{\displaystyle {\mathcal {L}}={\mathcal {L}}^{\downarrow }.}
A familyL{\displaystyle {\mathcal {L}}}is said to be:
A familyL{\displaystyle {\mathcal {L}}}of sets is called a/an:
Sequencesof sets often arise inmeasure theory.
Algebra of sets
AfamilyΦ{\displaystyle \Phi }of subsets of a setX{\displaystyle X}is said to bean algebra of setsif∅∈Φ{\displaystyle \varnothing \in \Phi }and for allL,R∈Φ,{\displaystyle L,R\in \Phi ,}all three of the setsX∖R,L∩R,{\displaystyle X\setminus R,\,L\cap R,}andL∪R{\displaystyle L\cup R}are elements ofΦ.{\displaystyle \Phi .}[13]Thearticle on this topiclists set identities and other relationships these three operations.
Every algebra of sets is also aring of sets[13]and aπ-system.
Algebra generated by a family of sets
Given any familyS{\displaystyle {\mathcal {S}}}of subsets ofX,{\displaystyle X,}there is a unique smallest[note 7]algebra of sets inX{\displaystyle X}containingS.{\displaystyle {\mathcal {S}}.}[13]It is calledthe algebra generated byS{\displaystyle {\mathcal {S}}}and it will be denote it byΦS.{\displaystyle \Phi _{\mathcal {S}}.}This algebra can be constructed as follows:[13]
LetL,M,{\displaystyle {\mathcal {L}},{\mathcal {M}},}andR{\displaystyle {\mathcal {R}}}be families of sets overX.{\displaystyle X.}On the left hand sides of the following identities,L{\displaystyle {\mathcal {L}}}is theLeft most family,M{\displaystyle {\mathcal {M}}}is in theMiddle, andR{\displaystyle {\mathcal {R}}}is theRight most set.
Commutativity:[12]L(∪)R=R(∪)L{\displaystyle {\mathcal {L}}\;(\cup )\;{\mathcal {R}}={\mathcal {R}}\;(\cup )\;{\mathcal {L}}}L(∩)R=R(∩)L{\displaystyle {\mathcal {L}}\;(\cap )\;{\mathcal {R}}={\mathcal {R}}\;(\cap )\;{\mathcal {L}}}
Associativity:[12][L(∪)M](∪)R=L(∪)[M(∪)R]{\displaystyle [{\mathcal {L}}\;(\cup )\;{\mathcal {M}}]\;(\cup )\;{\mathcal {R}}={\mathcal {L}}\;(\cup )\;[{\mathcal {M}}\;(\cup )\;{\mathcal {R}}]}[L(∩)M](∩)R=L(∩)[M(∩)R]{\displaystyle [{\mathcal {L}}\;(\cap )\;{\mathcal {M}}]\;(\cap )\;{\mathcal {R}}={\mathcal {L}}\;(\cap )\;[{\mathcal {M}}\;(\cap )\;{\mathcal {R}}]}
Identity:L(∪){∅}=L{\displaystyle {\mathcal {L}}\;(\cup )\;\{\varnothing \}={\mathcal {L}}}L(∩){X}=L{\displaystyle {\mathcal {L}}\;(\cap )\;\{X\}={\mathcal {L}}}L(∖){∅}=L{\displaystyle {\mathcal {L}}\;(\setminus )\;\{\varnothing \}={\mathcal {L}}}
Domination:L(∪){X}={X}ifL≠∅{\displaystyle {\mathcal {L}}\;(\cup )\;\{X\}=\{X\}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing }L(∩){∅}={∅}ifL≠∅{\displaystyle {\mathcal {L}}\;(\cap )\;\{\varnothing \}=\{\varnothing \}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing }L(∪)∅=∅{\displaystyle {\mathcal {L}}\;(\cup )\;\varnothing =\varnothing }L(∩)∅=∅{\displaystyle {\mathcal {L}}\;(\cap )\;\varnothing =\varnothing }L(∖)∅=∅{\displaystyle {\mathcal {L}}\;(\setminus )\;\varnothing =\varnothing }∅(∖)R=∅{\displaystyle \varnothing \;(\setminus )\;{\mathcal {R}}=\varnothing }
℘(L∩R)=℘(L)∩℘(R){\displaystyle \wp (L\cap R)~=~\wp (L)\cap \wp (R)}℘(L∪R)=℘(L)(∪)℘(R)⊇℘(L)∪℘(R).{\displaystyle \wp (L\cup R)~=~\wp (L)\ (\cup )\ \wp (R)~\supseteq ~\wp (L)\cup \wp (R).}
IfL{\displaystyle L}andR{\displaystyle R}are subsets of a vector spaceX{\displaystyle X}and ifs{\displaystyle s}is a scalar then℘(sL)=s℘(L){\displaystyle \wp (sL)~=~s\wp (L)}℘(L+R)⊇℘(L)+℘(R).{\displaystyle \wp (L+R)~\supseteq ~\wp (L)+\wp (R).}
Suppose thatL{\displaystyle L}is any set such thatL⊇Ri{\displaystyle L\supseteq R_{i}}for every indexi.{\displaystyle i.}IfR∙{\displaystyle R_{\bullet }}decreases toR{\displaystyle R}thenL∖R∙:=(L∖Ri)i{\displaystyle L\setminus R_{\bullet }:=\left(L\setminus R_{i}\right)_{i}}increases toL∖R{\displaystyle L\setminus R}[11]whereas if insteadR∙{\displaystyle R_{\bullet }}increases toR{\displaystyle R}thenL∖R∙{\displaystyle L\setminus R_{\bullet }}decreases toL∖R.{\displaystyle L\setminus R.}
IfLandR{\displaystyle L{\text{ and }}R}are arbitrary sets and ifL∙=(Li)i{\displaystyle L_{\bullet }=\left(L_{i}\right)_{i}}increases (resp. decreases) toL{\displaystyle L}then(Li∖R)i{\displaystyle \left(L_{i}\setminus R\right)_{i}}increase (resp. decreases) toL∖R.{\displaystyle L\setminus R.}
Suppose thatS∙=(Si)i=1∞{\displaystyle S_{\bullet }=\left(S_{i}\right)_{i=1}^{\infty }}is any sequence of sets, thatS⊆⋃iSi{\displaystyle S\subseteq \bigcup _{i}S_{i}}is any subset, and for every indexi,{\displaystyle i,}letDi=(Si∩S)∖⋃m=1i(Sm∩S).{\displaystyle D_{i}=\left(S_{i}\cap S\right)\setminus \bigcup _{m=1}^{i}\left(S_{m}\cap S\right).}ThenS=⋃iDi{\displaystyle S=\bigcup _{i}D_{i}}andD∙:=(Di)i=1∞{\displaystyle D_{\bullet }:=\left(D_{i}\right)_{i=1}^{\infty }}is a sequence of pairwise disjoint sets.[11]
Suppose thatS∙=(Si)i=1∞{\displaystyle S_{\bullet }=\left(S_{i}\right)_{i=1}^{\infty }}is non-decreasing, letS0=∅,{\displaystyle S_{0}=\varnothing ,}and letDi=Si∖Si−1{\displaystyle D_{i}=S_{i}\setminus S_{i-1}}for everyi=1,2,….{\displaystyle i=1,2,\ldots .}Then⋃iSi=⋃iDi{\displaystyle \bigcup _{i}S_{i}=\bigcup _{i}D_{i}}andD∙=(Di)i=1∞{\displaystyle D_{\bullet }=\left(D_{i}\right)_{i=1}^{\infty }}is a sequence of pairwise disjoint sets.[11]
Notes
Proofs
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https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
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Infunctional analysisand related areas ofmathematics, analmost open mapbetweentopological spacesis amapthat satisfies a condition similar to, but weaker than, the condition of being anopen map.
As described below, for certain broad categories oftopological vector spaces,allsurjectivelinear operators are necessarily almost open.
Given a surjective mapf:X→Y,{\displaystyle f:X\to Y,}a pointx∈X{\displaystyle x\in X}is called apoint of opennessforf{\displaystyle f}andf{\displaystyle f}is said to beopen atx{\displaystyle x}(oran open map atx{\displaystyle x}) if for every open neighborhoodU{\displaystyle U}ofx,{\displaystyle x,}f(U){\displaystyle f(U)}is aneighborhoodoff(x){\displaystyle f(x)}inY{\displaystyle Y}(note that the neighborhoodf(U){\displaystyle f(U)}is not required to be anopenneighborhood).
A surjective map is called anopen mapif it is open at every point of its domain, while it is called analmost open mapif each of itsfibershas some point of openness.
Explicitly, a surjective mapf:X→Y{\displaystyle f:X\to Y}is said to bealmost openif for everyy∈Y,{\displaystyle y\in Y,}there exists somex∈f−1(y){\displaystyle x\in f^{-1}(y)}such thatf{\displaystyle f}is open atx.{\displaystyle x.}Every almost open surjection is necessarily apseudo-open map(introduced byAlexander Arhangelskiiin 1963), which by definition means that for everyy∈Y{\displaystyle y\in Y}and every neighborhoodU{\displaystyle U}off−1(y){\displaystyle f^{-1}(y)}(that is,f−1(y)⊆IntXU{\displaystyle f^{-1}(y)\subseteq \operatorname {Int} _{X}U}),f(U){\displaystyle f(U)}is necessarily a neighborhood ofy.{\displaystyle y.}
A linear mapT:X→Y{\displaystyle T:X\to Y}between twotopological vector spaces(TVSs) is called anearly open linear mapor analmost open linear mapif for any neighborhoodU{\displaystyle U}of0{\displaystyle 0}inX,{\displaystyle X,}the closure ofT(U){\displaystyle T(U)}inY{\displaystyle Y}is a neighborhood of the origin.
Importantly, some authors use a different definition of "almost open map" in which they instead require that the linear mapT{\displaystyle T}satisfy: for any neighborhoodU{\displaystyle U}of0{\displaystyle 0}inX,{\displaystyle X,}the closure ofT(U){\displaystyle T(U)}inT(X){\displaystyle T(X)}(rather than inY{\displaystyle Y}) is a neighborhood of the origin;
this article will not use this definition.[1]
If a linear mapT:X→Y{\displaystyle T:X\to Y}is almost open then becauseT(X){\displaystyle T(X)}is a vector subspace ofY{\displaystyle Y}that contains a neighborhood of the origin inY,{\displaystyle Y,}the mapT:X→Y{\displaystyle T:X\to Y}is necessarilysurjective.
For this reason many authors require surjectivity as part of the definition of "almost open".
IfT:X→Y{\displaystyle T:X\to Y}is a bijective linear operator, thenT{\displaystyle T}is almost open if and only ifT−1{\displaystyle T^{-1}}isalmost continuous.[1]
Every surjectiveopen mapis an almost open map but in general, the converse is not necessarily true.
If a surjectionf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is an almost open map then it will be an open map if it satisfies the following condition (a condition that doesnotdepend in any way onY{\displaystyle Y}'s topologyσ{\displaystyle \sigma }):
If the map is continuous then the above condition is also necessary for the map to be open. That is, iff:X→Y{\displaystyle f:X\to Y}is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.
The two theorems above donotrequire the surjective linear map to satisfyanytopological conditions.
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Inmathematics, particularly infunctional analysisandtopology,closed graphis a property offunctions.[1][2]A functionf:X→Ybetweentopological spaceshas aclosed graphif itsgraphis aclosed subsetof theproduct spaceX×Y.
A related property isopen graph.[3]
This property is studied because there are many theorems, known asclosed graph theorems, giving conditions under which a function with a closed graph is necessarilycontinuous. One particularly well-known class of closed graph theorems are theclosed graph theorems in functional analysis.
We give the more general definition of when aY-valued function or set-valued function defined on asubsetSofXhas a closed graph since this generality is needed in the study ofclosed linear operatorsthat are defined on a dense subspaceSof atopological vector spaceX(and not necessarily defined on all ofX).
This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.
Note that we may define anopengraph, asequentially closedgraph, and a sequentially open graph in similar ways.
When reading literature infunctional analysis, iff:X→Yis a linear map between topological vector spaces (TVSs) (e.g.Banach spaces) then "fis closed" will almost always means the following:
Otherwise, especially in literature aboutpoint-set topology, "fis closed" may instead mean the following:
These two definitions of "closed map" are not equivalent.
If it is unclear, then it is recommended that a reader check how "closed map" is defined by the literature they are reading.
Throughout, letXandYbe topological spaces.
Iff:X→Yis a function then the following are equivalent:
and ifYis aHausdorff spacethat iscompact, then we may add to this list:
and if bothXandYarefirst-countablespaces then we may add to this list:
Iff:X→Yis a function then the following are equivalent:
IfF:X→ 2Yis a set-valued function between topological spacesXandYthen the following are equivalent:
and ifYis compact and Hausdorff then we may add to this list:
and if bothXandYare metrizable spaces then we may add to this list:
Throughout, letX{\displaystyle X}andY{\displaystyle Y}be topological spaces andX×Y{\displaystyle X\times Y}is endowed with the product topology.
Iff:X→Y{\displaystyle f:X\to Y}is a function then it is said to have aclosed graphif it satisfies any of the following are equivalent conditions:
and ifY{\displaystyle Y}is a Hausdorff compact space then we may add to this list:
and if bothX{\displaystyle X}andY{\displaystyle Y}arefirst-countablespaces then we may add to this list:
Function with a sequentially closed graph
Iff:X→Y{\displaystyle f:X\to Y}is a function then the following are equivalent:
Conditions that guarantee that a function with a closed graph is necessarily continuous are calledclosed graph theorems.
Closed graph theorems are of particular interest infunctional analysiswhere there are many theorems giving conditions under which alinear mapwith a closed graph is necessarily continuous.
For examples in functional analysis, seecontinuous linear operator.
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Infunctional analysis, a branch of mathematics, aclosed linear operatoror often aclosed operatoris alinear operatorwhose graph is closed (seeclosed graph property). It is a basic example of anunbounded operator.
Theclosed graph theoremsays a linear operatorf:X→Y{\displaystyle f:X\to Y}betweenBanach spacesis a closed operator if and only if it is abounded operatorand the domain of the operator isX{\displaystyle X}. Hence, a closed linear operator that is used in practice is typically onlydefined on a dense subspaceof a Banach space.
It is common in functional analysis to considerpartial functions, which are functions defined on asubsetof some spaceX.{\displaystyle X.}A partial functionf{\displaystyle f}is declared with the notationf:D⊆X→Y,{\displaystyle f:D\subseteq X\to Y,}which indicates thatf{\displaystyle f}has prototypef:D→Y{\displaystyle f:D\to Y}(that is, itsdomainisD{\displaystyle D}and itscodomainisY{\displaystyle Y})
Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, thegraphof a partial functionf{\displaystyle f}is the setgraph(f)={(x,f(x)):x∈domf}.{\displaystyle \operatorname {graph} {\!(f)}=\{(x,f(x)):x\in \operatorname {dom} f\}.}However, one exception to this is the definition of "closed graph". Apartialfunctionf:D⊆X→Y{\displaystyle f:D\subseteq X\to Y}is said to have aclosed graphifgraphf{\displaystyle \operatorname {graph} f}is a closed subset ofX×Y{\displaystyle X\times Y}in theproduct topology; importantly, note that the product space isX×Y{\displaystyle X\times Y}andnotD×Y=domf×Y{\displaystyle D\times Y=\operatorname {dom} f\times Y}as it was defined above for ordinary functions. In contrast, whenf:D→Y{\displaystyle f:D\to Y}is considered as an ordinary function (rather than as the partial functionf:D⊆X→Y{\displaystyle f:D\subseteq X\to Y}), then "having a closed graph" would instead mean thatgraphf{\displaystyle \operatorname {graph} f}is a closed subset ofD×Y.{\displaystyle D\times Y.}Ifgraphf{\displaystyle \operatorname {graph} f}is a closed subset ofX×Y{\displaystyle X\times Y}then it is also a closed subset ofdom(f)×Y{\displaystyle \operatorname {dom} (f)\times Y}although the converse is not guaranteed in general.
Definition: IfXandYaretopological vector spaces(TVSs) then we call alinear mapf:D(f) ⊆X→Yaclosed linear operatorif its graph is closed inX×Y.
A linear operatorf:D⊆X→Y{\displaystyle f:D\subseteq X\to Y}isclosableinX×Y{\displaystyle X\times Y}if there exists avector subspaceE⊆X{\displaystyle E\subseteq X}containingD{\displaystyle D}and a function (resp. multifunction)F:E→Y{\displaystyle F:E\to Y}whose graph is equal to the closure of the setgraphf{\displaystyle \operatorname {graph} f}inX×Y.{\displaystyle X\times Y.}Such anF{\displaystyle F}is called aclosure off{\displaystyle f}inX×Y{\displaystyle X\times Y}, is denoted byf¯,{\displaystyle {\overline {f}},}and necessarily extendsf.{\displaystyle f.}
Iff:D⊆X→Y{\displaystyle f:D\subseteq X\to Y}is a closable linear operator then acoreor anessential domainoff{\displaystyle f}is a subsetC⊆D{\displaystyle C\subseteq D}such that the closure inX×Y{\displaystyle X\times Y}of the graph of the restrictionf|C:C→Y{\displaystyle f{\big \vert }_{C}:C\to Y}off{\displaystyle f}toC{\displaystyle C}is equal to the closure of the graph off{\displaystyle f}inX×Y{\displaystyle X\times Y}(i.e. the closure ofgraphf{\displaystyle \operatorname {graph} f}inX×Y{\displaystyle X\times Y}is equal to the closure ofgraphf|C{\displaystyle \operatorname {graph} f{\big \vert }_{C}}inX×Y{\displaystyle X\times Y}).
A bounded operator is a closed operator. Here are examples of closed operators that are not bounded.
The following properties are easily checked for a linear operatorf:D(f) ⊆X→Ybetween Banach spaces:
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Inmathematics, more specificallytopology, alocal homeomorphismis afunctionbetweentopological spacesthat, intuitively, preserves local (though not necessarily global) structure.
Iff:X→Y{\displaystyle f:X\to Y}is a local homeomorphism,X{\displaystyle X}is said to be anétale spaceoverY.{\displaystyle Y.}Local homeomorphisms are used in the study ofsheaves. Typical examples of local homeomorphisms arecovering maps.
A topological spaceX{\displaystyle X}islocally homeomorphictoY{\displaystyle Y}if every point ofX{\displaystyle X}has a neighborhood that ishomeomorphicto an open subset ofY.{\displaystyle Y.}For example, amanifoldof dimensionn{\displaystyle n}is locally homeomorphic toRn.{\displaystyle \mathbb {R} ^{n}.}
If there is a local homeomorphism fromX{\displaystyle X}toY,{\displaystyle Y,}thenX{\displaystyle X}is locally homeomorphic toY,{\displaystyle Y,}but the converse is not always true.
For example, the two dimensionalsphere, being a manifold, is locally homeomorphic to the planeR2,{\displaystyle \mathbb {R} ^{2},}but there is no local homeomorphismS2→R2.{\displaystyle S^{2}\to \mathbb {R} ^{2}.}
A functionf:X→Y{\displaystyle f:X\to Y}between twotopological spacesis called alocal homeomorphism[1]if every pointx∈X{\displaystyle x\in X}has anopen neighborhoodU{\displaystyle U}whoseimagef(U){\displaystyle f(U)}is open inY{\displaystyle Y}and therestrictionf|U:U→f(U){\displaystyle f{\big \vert }_{U}:U\to f(U)}is ahomeomorphism(where the respectivesubspace topologiesare used onU{\displaystyle U}and onf(U){\displaystyle f(U)}).
Local homeomorphisms versus homeomorphisms
Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it isbijective.
A local homeomorphism need not be a homeomorphism. For example, the functionR→S1{\displaystyle \mathbb {R} \to S^{1}}defined byt↦eit{\displaystyle t\mapsto e^{it}}(so that geometrically, this map wraps thereal linearound thecircle) is a local homeomorphism but not a homeomorphism.
The mapf:S1→S1{\displaystyle f:S^{1}\to S^{1}}defined byf(z)=zn,{\displaystyle f(z)=z^{n},}which wraps the circle around itselfn{\displaystyle n}times (that is, haswinding numbern{\displaystyle n}), is a local homeomorphism for all non-zeron,{\displaystyle n,}but it is a homeomorphism only when it isbijective(that is, only whenn=1{\displaystyle n=1}orn=−1{\displaystyle n=-1}).
Generalizing the previous two examples, everycovering mapis a local homeomorphism; in particular, theuniversal coverp:C→Y{\displaystyle p:C\to Y}of a spaceY{\displaystyle Y}is a local homeomorphism.
In certain situations the converse is true. For example: ifp:X→Y{\displaystyle p:X\to Y}is aproperlocal homeomorphism between twoHausdorff spacesand ifY{\displaystyle Y}is alsolocally compact, thenp{\displaystyle p}is a covering map.
Local homeomorphisms and composition of functions
Thecompositionof two local homeomorphisms is a local homeomorphism; explicitly, iff:X→Y{\displaystyle f:X\to Y}andg:Y→Z{\displaystyle g:Y\to Z}are local homeomorphisms then the compositiong∘f:X→Z{\displaystyle g\circ f:X\to Z}is also a local homeomorphism.
The restriction of a local homeomorphism to any open subset of the domain will again be a local homomorphism; explicitly, iff:X→Y{\displaystyle f:X\to Y}is a local homeomorphism then its restrictionf|U:U→Y{\displaystyle f{\big \vert }_{U}:U\to Y}to anyU{\displaystyle U}open subset ofX{\displaystyle X}is also a local homeomorphism.
Iff:X→Y{\displaystyle f:X\to Y}is continuous while bothg:Y→Z{\displaystyle g:Y\to Z}andg∘f:X→Z{\displaystyle g\circ f:X\to Z}are local homeomorphisms, thenf{\displaystyle f}is also a local homeomorphism.
Inclusion maps
IfU⊆X{\displaystyle U\subseteq X}is any subspace (where as usual,U{\displaystyle U}is equipped with thesubspace topologyinduced byX{\displaystyle X}) then theinclusion mapi:U→X{\displaystyle i:U\to X}is always atopological embedding. But it is a local homeomorphism if and only ifU{\displaystyle U}is open inX.{\displaystyle X.}The subsetU{\displaystyle U}being open inX{\displaystyle X}is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset ofX{\displaystyle X}neveryields a local homeomorphism (since it will not be an open map).
The restrictionf|U:U→Y{\displaystyle f{\big \vert }_{U}:U\to Y}of a functionf:X→Y{\displaystyle f:X\to Y}to a subsetU⊆X{\displaystyle U\subseteq X}is equal to its composition with the inclusion mapi:U→X;{\displaystyle i:U\to X;}explicitly,f|U=f∘i.{\displaystyle f{\big \vert }_{U}=f\circ i.}Since the composition of two local homeomorphisms is a local homeomorphism, iff:X→Y{\displaystyle f:X\to Y}andi:U→X{\displaystyle i:U\to X}are local homomorphisms then so isf|U=f∘i.{\displaystyle f{\big \vert }_{U}=f\circ i.}Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms.
Invariance of domain
Invariance of domainguarantees that iff:U→Rn{\displaystyle f:U\to \mathbb {R} ^{n}}is acontinuousinjective mapfrom an open subsetU{\displaystyle U}ofRn,{\displaystyle \mathbb {R} ^{n},}thenf(U){\displaystyle f(U)}is open inRn{\displaystyle \mathbb {R} ^{n}}andf:U→f(U){\displaystyle f:U\to f(U)}is ahomeomorphism.
Consequently, a continuous mapf:U→Rn{\displaystyle f:U\to \mathbb {R} ^{n}}from an open subsetU⊆Rn{\displaystyle U\subseteq \mathbb {R} ^{n}}will be a local homeomorphism if and only if it is alocallyinjective map(meaning that every point inU{\displaystyle U}has aneighborhoodN{\displaystyle N}such that the restriction off{\displaystyle f}toN{\displaystyle N}is injective).
Local homeomorphisms in analysis
It is shown incomplex analysisthat a complexanalyticfunctionf:U→C{\displaystyle f:U\to \mathbb {C} }(whereU{\displaystyle U}is an open subset of thecomplex planeC{\displaystyle \mathbb {C} }) is a local homeomorphism precisely when thederivativef′(z){\displaystyle f^{\prime }(z)}is non-zero for allz∈U.{\displaystyle z\in U.}The functionf(x)=zn{\displaystyle f(x)=z^{n}}on an open disk around0{\displaystyle 0}is not a local homeomorphism at0{\displaystyle 0}whenn≥2.{\displaystyle n\geq 2.}In that case0{\displaystyle 0}is a point of "ramification" (intuitively,n{\displaystyle n}sheets come together there).
Using theinverse function theoremone can show that a continuously differentiable functionf:U→Rn{\displaystyle f:U\to \mathbb {R} ^{n}}(whereU{\displaystyle U}is an open subset ofRn{\displaystyle \mathbb {R} ^{n}}) is a local homeomorphism if the derivativeDxf{\displaystyle D_{x}f}is an invertible linear map (invertible square matrix) for everyx∈U.{\displaystyle x\in U.}(The converse is false, as shown by the local homeomorphismf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }withf(x)=x3{\displaystyle f(x)=x^{3}}).
An analogous condition can be formulated for maps betweendifferentiable manifolds.
Local homeomorphisms and fibers
Supposef:X→Y{\displaystyle f:X\to Y}is a continuousopensurjection between twoHausdorffsecond-countablespaces whereX{\displaystyle X}is aBaire spaceandY{\displaystyle Y}is anormal space. If everyfiberoff{\displaystyle f}is adiscrete subspaceofX{\displaystyle X}(which is a necessary condition forf:X→Y{\displaystyle f:X\to Y}to be a local homeomorphism) thenf{\displaystyle f}is aY{\displaystyle Y}-valued local homeomorphism on a dense open subset ofX.{\displaystyle X.}To clarify this statement's conclusion, letO=Of{\displaystyle O=O_{f}}be the (unique) largest open subset ofX{\displaystyle X}such thatf|O:O→Y{\displaystyle f{\big \vert }_{O}:O\to Y}is a local homeomorphism.[note 1]If everyfiberoff{\displaystyle f}is adiscrete subspaceofX{\displaystyle X}then this open setO{\displaystyle O}is necessarily adensesubsetofX.{\displaystyle X.}In particular, ifX≠∅{\displaystyle X\neq \varnothing }thenO≠∅;{\displaystyle O\neq \varnothing ;}a conclusion that may be false without the assumption thatf{\displaystyle f}'s fibers are discrete (see this footnote[note 2]for an example).
One corollary is that every continuous open surjectionf{\displaystyle f}betweencompletely metrizablesecond-countable spaces that hasdiscretefibers is "almost everywhere" a local homeomorphism (in the topological sense thatOf{\displaystyle O_{f}}is a dense open subset of its domain).
For example, the mapf:R→[0,∞){\displaystyle f:\mathbb {R} \to [0,\infty )}defined by the polynomialf(x)=x2{\displaystyle f(x)=x^{2}}is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subsetOf{\displaystyle O_{f}}is dense inR;{\displaystyle \mathbb {R} ;}with additional effort (using theinverse function theoremfor instance), it can be shown thatOf=R∖{0},{\displaystyle O_{f}=\mathbb {R} \setminus \{0\},}which confirms that this set is indeed dense inR.{\displaystyle \mathbb {R} .}This example also shows that it is possible forOf{\displaystyle O_{f}}to be aproperdense subset off{\displaystyle f}'s domain.
Becauseevery fiber of every non-constant polynomial is finite(and thus a discrete, and even compact, subspace), this example generalizes to such polynomials whenever the mapping induced by it is an open map.[note 3]
Local homeomorphisms and Hausdorffness
There exist local homeomorphismsf:X→Y{\displaystyle f:X\to Y}whereY{\displaystyle Y}is aHausdorff spacebutX{\displaystyle X}is not.
Consider for instance thequotient spaceX=(R⊔R)/∼,{\displaystyle X=\left(\mathbb {R} \sqcup \mathbb {R} \right)/\sim ,}where theequivalence relation∼{\displaystyle \sim }on thedisjoint unionof two copies of the reals identifies every negative real of the first copy with the corresponding negative real of the second copy.
The two copies of0{\displaystyle 0}are not identified and they do not have any disjoint neighborhoods, soX{\displaystyle X}is not Hausdorff. One readily checks that the natural mapf:X→R{\displaystyle f:X\to \mathbb {R} }is a local homeomorphism.
The fiberf−1({y}){\displaystyle f^{-1}(\{y\})}has two elements ify≥0{\displaystyle y\geq 0}and one element ify<0.{\displaystyle y<0.}Similarly, it is possible to construct a local homeomorphismsf:X→Y{\displaystyle f:X\to Y}whereX{\displaystyle X}is Hausdorff andY{\displaystyle Y}is not: pick the natural map fromX=R⊔R{\displaystyle X=\mathbb {R} \sqcup \mathbb {R} }toY=(R⊔R)/∼{\displaystyle Y=\left(\mathbb {R} \sqcup \mathbb {R} \right)/\sim }with the same equivalence relation∼{\displaystyle \sim }as above.
A map is a local homeomorphism if and only if it iscontinuous,open, andlocally injective. In particular, every local homeomorphism is a continuous andopen map. Abijectivelocal homeomorphism is therefore a homeomorphism.
Whether or not a functionf:X→Y{\displaystyle f:X\to Y}is a local homeomorphism depends on its codomain. Theimagef(X){\displaystyle f(X)}of a local homeomorphismf:X→Y{\displaystyle f:X\to Y}is necessarily an open subset of its codomainY{\displaystyle Y}andf:X→f(X){\displaystyle f:X\to f(X)}will also be a local homeomorphism (that is,f{\displaystyle f}will continue to be a local homeomorphism when it is considered as the surjective mapf:X→f(X){\displaystyle f:X\to f(X)}onto its image, wheref(X){\displaystyle f(X)}has thesubspace topologyinherited fromY{\displaystyle Y}). However, in general it is possible forf:X→f(X){\displaystyle f:X\to f(X)}to be a local homeomorphism butf:X→Y{\displaystyle f:X\to Y}tonotbe a local homeomorphism (as is the case with the mapf:R→R2{\displaystyle f:\mathbb {R} \to \mathbb {R} ^{2}}defined byf(x)=(x,0),{\displaystyle f(x)=(x,0),}for example). A mapf:X→Y{\displaystyle f:X\to Y}is a local homomorphism if and only iff:X→f(X){\displaystyle f:X\to f(X)}is a local homeomorphism andf(X){\displaystyle f(X)}is an open subset ofY.{\displaystyle Y.}
Everyfiberof a local homeomorphismf:X→Y{\displaystyle f:X\to Y}is adiscrete subspaceof itsdomainX.{\displaystyle X.}
A local homeomorphismf:X→Y{\displaystyle f:X\to Y}transfers "local" topological properties in both directions:
As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.
The local homeomorphisms withcodomainY{\displaystyle Y}stand in a natural one-to-one correspondence with thesheavesof sets onY;{\displaystyle Y;}this correspondence is in fact anequivalence of categories. Furthermore, every continuous map with codomainY{\displaystyle Y}gives rise to a uniquely defined local homeomorphism with codomainY{\displaystyle Y}in a natural way. All of this is explained in detail in the article onsheaves.
The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces.
Fordifferentiable manifolds, we obtain thelocal diffeomorphisms; forschemes, we have theformally étale morphismsand theétale morphisms; and fortoposes, we get theétale geometric morphisms.
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Intopologya branch of mathematics, aquasi-open maporquasi-interior mapis afunctionwhich has similar properties tocontinuous maps.
However, continuous maps and quasi-open maps are not related.[1]
A functionf:X→Ybetweentopological spacesXandYis quasi-open if, for any non-emptyopen setU⊆X, theinterioroff('U)inYis non-empty.[1][2]
Letf:X→Y{\displaystyle f:X\to Y}be a map betweentopological spaces.
Thistopology-relatedarticle is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Quasi-open_map
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Intopologyand related areas ofmathematics, thequotient spaceof atopological spaceunder a givenequivalence relationis a new topological space constructed by endowing thequotient setof the original topological space with thequotient topology, that is, with thefinest topologythat makescontinuousthecanonical projection map(the function that maps points to theirequivalence classes). In other words, a subset of a quotient space isopenif and only if itspreimageunder the canonical projection map is open in the original topological space.
Intuitively speaking, the points of each equivalence class areidentifiedor "glued together" for forming a new topological space. For example, identifying the points of aspherethat belong to the samediameterproduces theprojective planeas a quotient space.
LetX{\displaystyle X}be atopological space, and let∼{\displaystyle \sim }be anequivalence relationonX.{\displaystyle X.}Thequotient setY=X/∼{\displaystyle Y=X/{\sim }}is the set ofequivalence classesof elements ofX.{\displaystyle X.}The equivalence class ofx∈X{\displaystyle x\in X}is denoted[x].{\displaystyle [x].}
The construction ofY{\displaystyle Y}defines a canonicalsurjectionq:X∋x↦[x]∈Y.{\textstyle q:X\ni x\mapsto [x]\in Y.}As discussed below,q{\displaystyle q}is a quotient mapping, commonly called the canonical quotient map, or canonical projection map, associated toX/∼.{\displaystyle X/{\sim }.}
Thequotient spaceunder∼{\displaystyle \sim }is the setY{\displaystyle Y}equipped with thequotient topology, whoseopen setsare thosesubsetsU⊆Y{\textstyle U\subseteq Y}whosepreimageq−1(U){\displaystyle q^{-1}(U)}isopen. In other words,U{\displaystyle U}is open in the quotient topology onX/∼{\displaystyle X/{\sim }}if and only if{x∈X:[x]∈U}{\textstyle \{x\in X:[x]\in U\}}is open inX.{\displaystyle X.}Similarly, a subsetS⊆Y{\displaystyle S\subseteq Y}isclosedif and only if{x∈X:[x]∈S}{\displaystyle \{x\in X:[x]\in S\}}is closed inX.{\displaystyle X.}
The quotient topology is thefinal topologyon the quotient set, with respect to the mapx↦[x].{\displaystyle x\mapsto [x].}
A mapf:X→Y{\displaystyle f:X\to Y}is aquotient map(sometimes called anidentification map[1]) if it issurjectiveandY{\displaystyle Y}is equipped with thefinal topologyinduced byf.{\displaystyle f.}The latter condition admits two more-elementary formulations: a subsetV⊆Y{\displaystyle V\subseteq Y}is open (closed) if and only iff−1(V){\displaystyle f^{-1}(V)}is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map.
Saturated sets
A subsetS{\displaystyle S}ofX{\displaystyle X}is calledsaturated(with respect tof{\displaystyle f}) if it is of the formS=f−1(T){\displaystyle S=f^{-1}(T)}for some setT,{\displaystyle T,}which is true if and only iff−1(f(S))=S.{\displaystyle f^{-1}(f(S))=S.}The assignmentT↦f−1(T){\displaystyle T\mapsto f^{-1}(T)}establishes aone-to-one correspondence(whose inverse isS↦f(S){\displaystyle S\mapsto f(S)}) between subsetsT{\displaystyle T}ofY=f(X){\displaystyle Y=f(X)}and saturated subsets ofX.{\displaystyle X.}With this terminology, a surjectionf:X→Y{\displaystyle f:X\to Y}is a quotient map if and only if for everysaturatedsubsetS{\displaystyle S}ofX,{\displaystyle X,}S{\displaystyle S}is open inX{\displaystyle X}if and only iff(S){\displaystyle f(S)}is open inY.{\displaystyle Y.}In particular, open subsets ofX{\displaystyle X}that arenotsaturated have no impact on whether the functionf{\displaystyle f}is a quotient map (or, indeed, continuous: a functionf:X→Y{\displaystyle f:X\to Y}is continuous if and only if, for every saturatedS⊆X{\textstyle S\subseteq X}such thatf(S){\displaystyle f(S)}is open inf(X){\textstyle f(X)},the setS{\displaystyle S}is open inX{\textstyle X}).
Indeed, ifτ{\displaystyle \tau }is atopologyonX{\displaystyle X}andf:X→Y{\displaystyle f:X\to Y}is any map, then the setτf{\displaystyle \tau _{f}}of allU∈τ{\displaystyle U\in \tau }that are saturated subsets ofX{\displaystyle X}forms a topology onX.{\displaystyle X.}IfY{\displaystyle Y}is also a topological space thenf:(X,τ)→Y{\displaystyle f:(X,\tau )\to Y}is a quotient map (respectively,continuous) if and only if the same is true off:(X,τf)→Y.{\displaystyle f:\left(X,\tau _{f}\right)\to Y.}
Quotient space of fibers characterization
Given anequivalence relation∼{\displaystyle \,\sim \,}onX,{\displaystyle X,}denote theequivalence classof a pointx∈X{\displaystyle x\in X}by[x]:={z∈X:z∼x}{\displaystyle [x]:=\{z\in X:z\sim x\}}and letX/∼:={[x]:x∈X}{\displaystyle X/{\sim }:=\{[x]:x\in X\}}denote the set of equivalence classes. The mapq:X→X/∼{\displaystyle q:X\to X/{\sim }}that sends points to theirequivalence classes(that is, it is defined byq(x):=[x]{\displaystyle q(x):=[x]}for everyx∈X{\displaystyle x\in X}) is calledthe canonical map. It is asurjective mapand for alla,b∈X,{\displaystyle a,b\in X,}a∼b{\displaystyle a\,\sim \,b}if and only ifq(a)=q(b);{\displaystyle q(a)=q(b);}consequently,q(x)=q−1(q(x)){\displaystyle q(x)=q^{-1}(q(x))}for allx∈X.{\displaystyle x\in X.}In particular, this shows that the set of equivalence classX/∼{\displaystyle X/{\sim }}is exactly the set of fibers of the canonical mapq.{\displaystyle q.}IfX{\displaystyle X}is a topological space then givingX/∼{\displaystyle X/{\sim }}the quotient topology induced byq{\displaystyle q}will make it into a quotient space and makeq:X→X/∼{\displaystyle q:X\to X/{\sim }}into a quotient map.Up toahomeomorphism, this construction is representative of all quotient spaces; the precise meaning of this is now explained.
Letf:X→Y{\displaystyle f:X\to Y}be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for alla,b∈X{\displaystyle a,b\in X}thata∼b{\displaystyle a\,\sim \,b}if and only iff(a)=f(b).{\displaystyle f(a)=f(b).}Then∼{\displaystyle \,\sim \,}is an equivalence relation onX{\displaystyle X}such that for everyx∈X,{\displaystyle x\in X,}[x]=f−1(f(x)),{\displaystyle [x]=f^{-1}(f(x)),}which implies thatf([x]){\displaystyle f([x])}(defined byf([x])={f(z):z∈[x]}{\displaystyle f([x])=\{\,f(z)\,:z\in [x]\}}) is asingleton set; denote the unique element inf([x]){\displaystyle f([x])}byf^([x]){\displaystyle {\hat {f}}([x])}(so by definition,f([x])={f^([x])}{\displaystyle f([x])=\{\,{\hat {f}}([x])\,\}}).
The assignment[x]↦f^([x]){\displaystyle [x]\mapsto {\hat {f}}([x])}defines abijectionf^:X/∼→Y{\displaystyle {\hat {f}}:X/{\sim }\;\to \;Y}between the fibers off{\displaystyle f}and points inY.{\displaystyle Y.}Define the mapq:X→X/∼{\displaystyle q:X\to X/{\sim }}as above (byq(x):=[x]{\displaystyle q(x):=[x]}) and giveX/∼{\displaystyle X/{\sim }}the quotient topology induced byq{\displaystyle q}(which makesq{\displaystyle q}a quotient map). These maps are related by:f=f^∘qandq=f^−1∘f.{\displaystyle f={\hat {f}}\circ q\quad {\text{ and }}\quad q={\hat {f}}^{-1}\circ f.}From this and the fact thatq:X→X/∼{\displaystyle q:X\to X/{\sim }}is a quotient map, it follows thatf:X→Y{\displaystyle f:X\to Y}is continuous if and only if this is true off^:X/∼→Y.{\displaystyle {\hat {f}}:X/{\sim }\;\to \;Y.}Furthermore,f:X→Y{\displaystyle f:X\to Y}is a quotient map if and only iff^:X/∼→Y{\displaystyle {\hat {f}}:X/{\sim }\;\to \;Y}is ahomeomorphism(or equivalently, if and only if bothf^{\displaystyle {\hat {f}}}and its inverse are continuous).
Ahereditarily quotient mapis a surjective mapf:X→Y{\displaystyle f:X\to Y}with the property that for every subsetT⊆Y,{\displaystyle T\subseteq Y,}the restrictionf|f−1(T):f−1(T)→T{\displaystyle f{\big \vert }_{f^{-1}(T)}~:~f^{-1}(T)\to T}is also a quotient map.
There exist quotient maps that are not hereditarily quotient.
Quotient mapsq:X→Y{\displaystyle q:X\to Y}are characterized among surjective maps by the following property: ifZ{\displaystyle Z}is any topological space andf:Y→Z{\displaystyle f:Y\to Z}is any function, thenf{\displaystyle f}is continuous if and only iff∘q{\displaystyle f\circ q}is continuous.
The quotient spaceX/∼{\displaystyle X/{\sim }}together with the quotient mapq:X→X/∼{\displaystyle q:X\to X/{\sim }}is characterized by the followinguniversal property: ifg:X→Z{\displaystyle g:X\to Z}is a continuous map such thata∼b{\displaystyle a\sim b}impliesg(a)=g(b){\displaystyle g(a)=g(b)}for alla,b∈X,{\displaystyle a,b\in X,}then there exists a unique continuous mapf:X/∼→Z{\displaystyle f:X/{\sim }\to Z}such thatg=f∘q.{\displaystyle g=f\circ q.}In other words, the following diagram commutes:
One says thatg{\displaystyle g}descends to the quotientfor expressing this, that is that it factorizes through the quotient space. The continuous maps defined onX/∼{\displaystyle X/{\sim }}are, therefore, precisely those maps which arise from continuous maps defined onX{\displaystyle X}that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is copiously used when studying quotient spaces.
Given a continuous surjectionq:X→Y{\displaystyle q:X\to Y}it is useful to have criteria by which one can determine ifq{\displaystyle q}is a quotient map. Two sufficient criteria are thatq{\displaystyle q}beopenorclosed. Note that these conditions are onlysufficient, notnecessary. It is easy to construct examples of quotient maps that are neither open nor closed. For topological groups, the quotient map is open.
Separation
Connectedness
Compactness
Dimension
Topology
Algebra
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https://en.wikipedia.org/wiki/Quotient_map_(topology)
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Inmathematics, especiallytopology, aperfect mapis a particular kind ofcontinuous functionbetweentopological spaces. Perfect maps are weaker thanhomeomorphisms, but strong enough to preserve some topological properties such aslocal compactnessthat are not always preserved by continuous maps.
LetX{\displaystyle X}andY{\displaystyle Y}betopological spacesand letp{\displaystyle p}be a map fromX{\displaystyle X}toY{\displaystyle Y}that iscontinuous,closed,surjectiveand such that eachfiberp−1(y){\displaystyle p^{-1}(y)}iscompactrelative toX{\displaystyle X}for eachy{\displaystyle y}inY{\displaystyle Y}. Thenp{\displaystyle p}is known as a perfect map.
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https://en.wikipedia.org/wiki/Perfect_map
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Inmathematics, afunctionbetweentopological spacesis calledproperifinverse imagesofcompact subsetsare compact.[1]Inalgebraic geometry, theanalogousconcept is called aproper morphism.
There are several competing definitions of a "properfunction".
Some authors call a functionf:X→Y{\displaystyle f:X\to Y}between twotopological spacesproperif thepreimageof everycompactset inY{\displaystyle Y}is compact inX.{\displaystyle X.}Other authors call a mapf{\displaystyle f}properif it is continuous andclosed with compact fibers; that is if it is acontinuousclosed mapand the preimage of every point inY{\displaystyle Y}iscompact. The two definitions are equivalent ifY{\displaystyle Y}islocally compactandHausdorff.
Letf:X→Y{\displaystyle f:X\to Y}be a closed map, such thatf−1(y){\displaystyle f^{-1}(y)}is compact (inX{\displaystyle X}) for ally∈Y.{\displaystyle y\in Y.}LetK{\displaystyle K}be a compact subset ofY.{\displaystyle Y.}It remains to show thatf−1(K){\displaystyle f^{-1}(K)}is compact.
Let{Ua:a∈A}{\displaystyle \left\{U_{a}:a\in A\right\}}be an open cover off−1(K).{\displaystyle f^{-1}(K).}Then for allk∈K{\displaystyle k\in K}this is also an open cover off−1(k).{\displaystyle f^{-1}(k).}Since the latter is assumed to be compact, it has a finite subcover. In other words, for everyk∈K,{\displaystyle k\in K,}there exists a finite subsetγk⊆A{\displaystyle \gamma _{k}\subseteq A}such thatf−1(k)⊆∪a∈γkUa.{\displaystyle f^{-1}(k)\subseteq \cup _{a\in \gamma _{k}}U_{a}.}The setX∖∪a∈γkUa{\displaystyle X\setminus \cup _{a\in \gamma _{k}}U_{a}}is closed inX{\displaystyle X}and its image underf{\displaystyle f}is closed inY{\displaystyle Y}becausef{\displaystyle f}is a closed map. Hence the setVk=Y∖f(X∖∪a∈γkUa){\displaystyle V_{k}=Y\setminus f\left(X\setminus \cup _{a\in \gamma _{k}}U_{a}\right)}is open inY.{\displaystyle Y.}It follows thatVk{\displaystyle V_{k}}contains the pointk.{\displaystyle k.}NowK⊆∪k∈KVk{\displaystyle K\subseteq \cup _{k\in K}V_{k}}and becauseK{\displaystyle K}is assumed to be compact, there are finitely many pointsk1,…,ks{\displaystyle k_{1},\dots ,k_{s}}such thatK⊆∪i=1sVki.{\displaystyle K\subseteq \cup _{i=1}^{s}V_{k_{i}}.}Furthermore, the setΓ=∪i=1sγki{\displaystyle \Gamma =\cup _{i=1}^{s}\gamma _{k_{i}}}is a finite union of finite sets, which makesΓ{\displaystyle \Gamma }a finite set.
Now it follows thatf−1(K)⊆f−1(∪i=1sVki)⊆∪a∈ΓUa{\displaystyle f^{-1}(K)\subseteq f^{-1}\left(\cup _{i=1}^{s}V_{k_{i}}\right)\subseteq \cup _{a\in \Gamma }U_{a}}and we have found a finite subcover off−1(K),{\displaystyle f^{-1}(K),}which completes the proof.
IfX{\displaystyle X}is Hausdorff andY{\displaystyle Y}is locally compact Hausdorff then proper is equivalent touniversally closed. A map is universally closed if for any topological spaceZ{\displaystyle Z}the mapf×idZ:X×Z→Y×Z{\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z}is closed. In the case thatY{\displaystyle Y}is Hausdorff, this is equivalent to requiring that for any mapZ→Y{\displaystyle Z\to Y}the pullbackX×YZ→Z{\displaystyle X\times _{Y}Z\to Z}be closed, as follows from the fact thatX×YZ{\displaystyle X\times _{Y}Z}is a closed subspace ofX×Z.{\displaystyle X\times Z.}
An equivalent, possibly more intuitive definition whenX{\displaystyle X}andY{\displaystyle Y}aremetric spacesis as follows: we say an infinite sequence of points{pi}{\displaystyle \{p_{i}\}}in a topological spaceX{\displaystyle X}escapes to infinityif, for every compact setS⊆X{\displaystyle S\subseteq X}only finitely many pointspi{\displaystyle p_{i}}are inS.{\displaystyle S.}Then a continuous mapf:X→Y{\displaystyle f:X\to Y}is proper if and only if for every sequence of points{pi}{\displaystyle \left\{p_{i}\right\}}that escapes to infinity inX,{\displaystyle X,}the sequence{f(pi)}{\displaystyle \left\{f\left(p_{i}\right)\right\}}escapes to infinity inY.{\displaystyle Y.}
It is possible to generalize
the notion of proper maps of topological spaces tolocalesandtopoi, see (Johnstone 2002).
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https://en.wikipedia.org/wiki/Proper_map
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Inmathematics, specificallytopology, asequence covering mapis any of a class ofmapsbetweentopological spaceswhose definitions all somehow relate sequences in thecodomainwith sequences in thedomain. Examples includesequentiallyquotientmaps,sequence coverings,1-sequence coverings, and2-sequence coverings.[1][2][3][4]These classes of maps are closely related tosequential spaces. If the domain and/or codomain have certain additionaltopological properties(often, the spaces beingHausdorffandfirst-countableis more than enough) then these definitions become equivalent to other well-known classes of maps, such asopen mapsorquotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms ofsequential continuityor the characterization of compactness in terms ofsequential compactness(whenever such characterizations hold).
A subsetS{\displaystyle S}of(X,τ){\displaystyle (X,\tau )}is said to besequentially openin(X,τ){\displaystyle (X,\tau )}if whenever a sequence inX{\displaystyle X}converges (in(X,τ){\displaystyle (X,\tau )}) to some point that belongs toS,{\displaystyle S,}then that sequence is necessarilyeventuallyinS{\displaystyle S}(i.e. at most finitely many points in the sequence do not belong toS{\displaystyle S}). The setSeqOpen(X,τ){\displaystyle \operatorname {SeqOpen} (X,\tau )}of all sequentially open subsets of(X,τ){\displaystyle (X,\tau )}forms atopologyonX{\displaystyle X}that isfiner thanX{\displaystyle X}'s given topologyτ.{\displaystyle \tau .}By definition,(X,τ){\displaystyle (X,\tau )}is called asequential spaceifτ=SeqOpen(X,τ).{\displaystyle \tau =\operatorname {SeqOpen} (X,\tau ).}Given a sequencex∙{\displaystyle x_{\bullet }}inX{\displaystyle X}and a pointx∈X,{\displaystyle x\in X,}x∙→x{\displaystyle x_{\bullet }\to x}in(X,τ){\displaystyle (X,\tau )}if and only ifx∙→x{\displaystyle x_{\bullet }\to x}in(X,SeqOpen(X,τ)).{\displaystyle (X,\operatorname {SeqOpen} (X,\tau )).}Moreover,SeqOpen(X,τ){\displaystyle \operatorname {SeqOpen} (X,\tau )}is thefinesttopology onX{\displaystyle X}for which this characterization of sequence convergence in(X,τ){\displaystyle (X,\tau )}holds.
A mapf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is calledsequentially continuousiff:(X,SeqOpen(X,τ))→(Y,SeqOpen(Y,σ)){\displaystyle f:(X,\operatorname {SeqOpen} (X,\tau ))\to (Y,\operatorname {SeqOpen} (Y,\sigma ))}iscontinuous, which happens if and only if for every sequencex∙=(xi)i=1∞{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}inX{\displaystyle X}and everyx∈X,{\displaystyle x\in X,}ifx∙→x{\displaystyle x_{\bullet }\to x}in(X,τ){\displaystyle (X,\tau )}then necessarilyf(x∙)→f(x){\displaystyle f\left(x_{\bullet }\right)\to f(x)}in(Y,σ).{\displaystyle (Y,\sigma ).}Every continuous map is sequentially continuous although in general, the converse may fail to hold.
In fact, a space(X,τ){\displaystyle (X,\tau )}is a sequential space if and only if it has the followinguniversal propertyfor sequential spaces:
Thesequential closurein(X,τ){\displaystyle (X,\tau )}of a subsetS⊆X{\displaystyle S\subseteq X}is the setscl(X,τ)S{\displaystyle \operatorname {scl} _{(X,\tau )}S}consisting of allx∈X{\displaystyle x\in X}for which there exists a sequence inS{\displaystyle S}that converges tox{\displaystyle x}in(X,τ).{\displaystyle (X,\tau ).}A subsetS⊆X{\displaystyle S\subseteq X}is calledsequentially closedin(X,τ){\displaystyle (X,\tau )}ifS=scl(X,τ)S,{\displaystyle S=\operatorname {scl} _{(X,\tau )}S,}which happens if and only if whenever a sequence inS{\displaystyle S}converges in(X,τ){\displaystyle (X,\tau )}to some pointx∈X{\displaystyle x\in X}then necessarilyx∈S.{\displaystyle x\in S.}The space(X,τ){\displaystyle (X,\tau )}is called aFréchet–Urysohn spaceifsclXS=clXS{\displaystyle \operatorname {scl} _{X}S~=~\operatorname {cl} _{X}S}for every subsetS⊆X,{\displaystyle S\subseteq X,}which happens if and only if every subspace of(X,τ){\displaystyle (X,\tau )}is a sequential space.
Everyfirst-countable spaceis a Fréchet–Urysohn space and thus also a sequential space. Allpseudometrizable spaces,metrizable spaces, andsecond-countable spacesare first-countable.
Asequencex∙=(xi)i=1∞{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}in a setX{\displaystyle X}is by definition afunctionx∙:N→X{\displaystyle x_{\bullet }:\mathbb {N} \to X}whose value ati∈N{\displaystyle i\in \mathbb {N} }is denoted byxi{\displaystyle x_{i}}(although the usual notation used with functions, such as parenthesesx∙(i){\displaystyle x_{\bullet }(i)}orcompositionf∘x∙,{\displaystyle f\circ x_{\bullet },}might be used in certain situations to improve readability).
Statements such as "the sequencex∙{\displaystyle x_{\bullet }}isinjective" or "theimage(i.e. range)Imx∙{\displaystyle \operatorname {Im} x_{\bullet }}of a sequencex∙{\displaystyle x_{\bullet }}is infinite" as well as other terminology and notation that is defined for functions can thus be applied to sequences.
A sequences∙{\displaystyle s_{\bullet }}is said to be asubsequenceof another sequencex∙{\displaystyle x_{\bullet }}if there exists a strictly increasing mapl∙:N→N{\displaystyle l_{\bullet }:\mathbb {N} \to \mathbb {N} }(possibly denoted byl∙=(lk)k=1∞{\displaystyle l_{\bullet }=\left(l_{k}\right)_{k=1}^{\infty }}instead) such thatsk=xlk{\displaystyle s_{k}=x_{l_{k}}}for everyk∈N,{\displaystyle k\in \mathbb {N} ,}where this condition can be expressed in terms offunction composition∘{\displaystyle \circ }as:s∙=x∙∘l∙.{\displaystyle s_{\bullet }=x_{\bullet }\circ l_{\bullet }.}As usual, ifxl∙=(xlk)k=1∞{\displaystyle x_{l_{\bullet }}=\left(x_{l_{k}}\right)_{k=1}^{\infty }}is declared to be (such as by definition) a subsequence ofx∙{\displaystyle x_{\bullet }}then it should immediately be assumed thatl∙:N→N{\displaystyle l_{\bullet }:\mathbb {N} \to \mathbb {N} }is strictly increasing.
The notationx∙⊆S{\displaystyle x_{\bullet }\subseteq S}andImx∙⊆S{\displaystyle \operatorname {Im} x_{\bullet }\subseteq S}mean that the sequencex∙{\displaystyle x_{\bullet }}is valued in the setS.{\displaystyle S.}
The functionf:X→Y{\displaystyle f:X\to Y}is called asequence coveringif for every convergent sequencey∙{\displaystyle y_{\bullet }}inY,{\displaystyle Y,}there exists a sequencex∙⊆X{\displaystyle x_{\bullet }\subseteq X}such thaty∙=f∘x∙.{\displaystyle y_{\bullet }=f\circ x_{\bullet }.}It is called a1-sequence coveringif for everyy∈Y{\displaystyle y\in Y}there exists somex∈f−1(y){\displaystyle x\in f^{-1}(y)}such that every sequencey∙⊆Y{\displaystyle y_{\bullet }\subseteq Y}that converges toy{\displaystyle y}in(Y,σ),{\displaystyle (Y,\sigma ),}there exists a sequencex∙⊆X{\displaystyle x_{\bullet }\subseteq X}such thaty∙=f∘x∙{\displaystyle y_{\bullet }=f\circ x_{\bullet }}andx∙{\displaystyle x_{\bullet }}converges tox{\displaystyle x}in(X,τ).{\displaystyle (X,\tau ).}It is a2-sequence coveringiff:X→Y{\displaystyle f:X\to Y}is surjective and also for everyy∈Y{\displaystyle y\in Y}and everyx∈f−1(y),{\displaystyle x\in f^{-1}(y),}every sequencey∙⊆Y{\displaystyle y_{\bullet }\subseteq Y}and converges toy{\displaystyle y}in(Y,σ),{\displaystyle (Y,\sigma ),}there exists a sequencex∙⊆X{\displaystyle x_{\bullet }\subseteq X}such thaty∙=f∘x∙{\displaystyle y_{\bullet }=f\circ x_{\bullet }}andx∙{\displaystyle x_{\bullet }}converges tox{\displaystyle x}in(X,τ).{\displaystyle (X,\tau ).}A mapf:X→Y{\displaystyle f:X\to Y}is acompact coveringif for every compactK⊆Y{\displaystyle K\subseteq Y}there exists some compact subsetC⊆X{\displaystyle C\subseteq X}such thatf(C)=K.{\displaystyle f(C)=K.}
In analogy with the definition of sequential continuity, a mapf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is called asequentially quotient mapif
is aquotient map,[5]which happens if and only if for any subsetS⊆Y,{\displaystyle S\subseteq Y,}S{\displaystyle S}is sequentially open(Y,σ){\displaystyle (Y,\sigma )}if and only if this is true off−1(S){\displaystyle f^{-1}(S)}in(X,τ).{\displaystyle (X,\tau ).}Sequentially quotient maps were introduced inBoone & Siwiec 1976who defined them as above.[5]
Every sequentially quotient map is necessarily surjective and sequentially continuous although they may fail to be continuous.
Iff:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is a sequentially continuous surjection whose domain(X,τ){\displaystyle (X,\tau )}is asequential space, thenf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is aquotient mapif and only if(Y,σ){\displaystyle (Y,\sigma )}is a sequential space andf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is a sequentially quotient map.
Call a space(Y,σ){\displaystyle (Y,\sigma )}sequentially Hausdorffif(Y,SeqOpen(Y,σ)){\displaystyle (Y,\operatorname {SeqOpen} (Y,\sigma ))}is aHausdorff space.[6]In an analogous manner, a "sequential version" of every otherseparation axiomcan be defined in terms of whether or not the space(Y,SeqOpen(Y,σ)){\displaystyle (Y,\operatorname {SeqOpen} (Y,\sigma ))}possess it.
Every Hausdorff space is necessarily sequentially Hausdorff. A sequential space is Hausdorff if and only if it is sequentially Hausdorff.
Iff:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is a sequentially continuous surjection then assuming that(Y,σ){\displaystyle (Y,\sigma )}is sequentially Hausdorff, the following are equivalent:
If the assumption thatY{\displaystyle Y}is sequentially Hausdorff were to be removed, then statement (2) would still imply the other two statement but the above characterization would no longer be guaranteed to hold (however, if points in the codomain were required to be sequentially closed then any sequentially quotient map would necessarily satisfy condition (3)).
This remains true even if the sequential continuity requirement onf:X→Y{\displaystyle f:X\to Y}was strengthened to require (ordinary) continuity.
Instead of using the original definition, some authors define "sequentially quotient map" to mean acontinuoussurjection that satisfies condition (2) or alternatively, condition (3). If the codomain is sequentially Hausdorff then these definitions differs from the originalonlyin the added requirement of continuity (rather than merely requiring sequential continuity).
The mapf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is calledpresequentialif for every convergent sequencey∙→y{\displaystyle y_{\bullet }\to y}in(Y,σ){\displaystyle (Y,\sigma )}such thaty∙{\displaystyle y_{\bullet }}is not eventually equal toy,{\displaystyle y,}the set⋃yi≠yi∈N,f−1(yi){\displaystyle \bigcup _{\stackrel {i\in \mathbb {N} ,}{y_{i}\neq y}}f^{-1}\left(y_{i}\right)}isnotsequentially closed in(X,τ),{\displaystyle (X,\tau ),}[5]where this set may also be described as:
Equivalently,f:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is presequential if and only if for every convergent sequencey∙→y{\displaystyle y_{\bullet }\to y}in(Y,σ){\displaystyle (Y,\sigma )}such thaty∙⊆Y∖{y},{\displaystyle y_{\bullet }\subseteq Y\setminus \{y\},}the setf−1(Imy∙){\displaystyle f^{-1}\left(\operatorname {Im} y_{\bullet }\right)}isnotsequentially closed in(X,τ).{\displaystyle (X,\tau ).}
A surjective mapf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}between Hausdorff spaces is sequentially quotient if and only if it is sequentially continuous and a presequential map.[5]
Iff:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is a continuous surjection between twofirst-countableHausdorffspaces then the following statements are true:[7][8][9][10][11][12][3][4]
and if in addition bothX{\displaystyle X}andY{\displaystyle Y}areseparablemetric spacesthen to this list may be appended:
The following is a sufficient condition for a continuous surjection to be sequentially open, which with additional assumptions, results in a characterization ofopen maps. Assume thatf:X→Y{\displaystyle f:X\to Y}is a continuous surjection from aregular spaceX{\displaystyle X}onto a Hausdorff spaceY.{\displaystyle Y.}If the restrictionf|U:U→f(U){\displaystyle f{\big \vert }_{U}:U\to f(U)}is sequentially quotient for every open subsetU{\displaystyle U}ofX{\displaystyle X}thenf:X→Y{\displaystyle f:X\to Y}maps open subsets ofX{\displaystyle X}tosequentially opensubsets ofY.{\displaystyle Y.}Consequently, ifX{\displaystyle X}andY{\displaystyle Y}are alsosequential spaces, thenf:X→Y{\displaystyle f:X\to Y}is anopen mapif and only iff|U:U→f(U){\displaystyle f{\big \vert }_{U}:U\to f(U)}is sequentially quotient (or equivalently,quotient) for every open subsetU{\displaystyle U}ofX.{\displaystyle X.}
Given an elementy∈Y{\displaystyle y\in Y}in the codomain of a (not necessarily surjective) continuous functionf:X→Y,{\displaystyle f:X\to Y,}the following gives a sufficient condition fory{\displaystyle y}to belong tof{\displaystyle f}'s image:y∈Imf:=f(X).{\displaystyle y\in \operatorname {Im} f:=f(X).}AfamilyB{\displaystyle {\mathcal {B}}}of subsets of a topological space(X,τ){\displaystyle (X,\tau )}is said to belocally finiteat a pointx∈X{\displaystyle x\in X}if there exists some open neighborhoodU{\displaystyle U}ofx{\displaystyle x}such that the set{B∈B:U∩B≠∅}{\displaystyle \left\{B\in {\mathcal {B}}~:~U\cap B\neq \varnothing \right\}}is finite.
Assume thatf:X→Y{\displaystyle f:X\to Y}is a continuous map between twoHausdorfffirst-countable spacesand lety∈Y.{\displaystyle y\in Y.}If there exists a sequencey∙=(yi)i=1∞{\displaystyle y_{\bullet }=\left(y_{i}\right)_{i=1}^{\infty }}inY{\displaystyle Y}such that (1)y∙→y{\displaystyle y_{\bullet }\to y}and (2) there exists somex∈X{\displaystyle x\in X}such that{f−1(yi):i∈N}{\displaystyle \left\{f^{-1}\left(y_{i}\right)~:~i\in \mathbb {N} \right\}}isnotlocally finite atx,{\displaystyle x,}theny∈Imf=f(X).{\displaystyle y\in \operatorname {Im} f=f(X).}The converse is true if there is no point at whichf{\displaystyle f}islocally constant; that is, if there does not exist any non-empty open subset ofX{\displaystyle X}on whichf{\displaystyle f}restrictsto a constant map.
Supposef:X→Y{\displaystyle f:X\to Y}is a continuous open surjection from afirst-countable spaceX{\displaystyle X}onto aHausdorff spaceY,{\displaystyle Y,}letD⊆Y{\displaystyle D\subseteq Y}be any non-empty subset, and lety∈clYD{\displaystyle y\in \operatorname {cl} _{Y}D}whereclYD{\displaystyle \operatorname {cl} _{Y}D}denotes the closure ofD{\displaystyle D}inY.{\displaystyle Y.}Then given anyx,z∈f−1(y){\displaystyle x,z\in f^{-1}(y)}and any sequencex∙{\displaystyle x_{\bullet }}inf−1(D){\displaystyle f^{-1}(D)}that converges tox,{\displaystyle x,}there exists a sequencez∙{\displaystyle z_{\bullet }}inf−1(D){\displaystyle f^{-1}(D)}that converges toz{\displaystyle z}as well as a subsequence(xlk)k=1∞{\displaystyle \left(x_{l_{k}}\right)_{k=1}^{\infty }}ofx∙{\displaystyle x_{\bullet }}such thatf(zk)=f(xlk){\displaystyle f(z_{k})=f\left(x_{l_{k}}\right)}for allk∈N.{\displaystyle k\in \mathbb {N} .}In short, this states that given a convergent sequencex∙⊆f−1(D){\displaystyle x_{\bullet }\subseteq f^{-1}(D)}such thatx∙→x{\displaystyle x_{\bullet }\to x}then for any otherz∈f−1(f(x)){\displaystyle z\in f^{-1}(f(x))}belonging to the same fiber asx,{\displaystyle x,}it is always possible to find a subsequencexl∙=(xlk)k=1∞{\displaystyle x_{l_{\bullet }}=\left(x_{l_{k}}\right)_{k=1}^{\infty }}such thatf∘xl∙=(f(xlk))k=1∞{\displaystyle f\circ x_{l_{\bullet }}=\left(f\left(x_{l_{k}}\right)\right)_{k=1}^{\infty }}can be "lifted" byf{\displaystyle f}to a sequence that converges toz.{\displaystyle z.}
The following shows that under certain conditions, a map'sfiberbeing acountable setis enough to guarantee the existence of apoint of openness. Iff:X→Y{\displaystyle f:X\to Y}is a sequence covering from a Hausdorffsequential spaceX{\displaystyle X}onto a Hausdorfffirst-countable spaceY{\displaystyle Y}and ify∈Y{\displaystyle y\in Y}is such that thefiberf−1(y){\displaystyle f^{-1}(y)}is a countable set, then there exists somex∈f−1(y){\displaystyle x\in f^{-1}(y)}such thatx{\displaystyle x}is a point of openness forf:X→Y.{\displaystyle f:X\to Y.}Consequently, iff:X→Y{\displaystyle f:X\to Y}isquotient mapbetween two Hausdorfffirst-countable spacesand if every fiber off{\displaystyle f}is countable, thenf:X→Y{\displaystyle f:X\to Y}is an almost open map and consequently, also a 1-sequence covering.
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Inmathematics, especiallyseveral complex variables, ananalytic polyhedronis a subset of thecomplex spaceCnof the form
whereDis a bounded connected open subset ofCn,fj{\displaystyle f_{j}}areholomorphiconDandPis assumed to berelatively compactinD.[1]Iffj{\displaystyle f_{j}}above are polynomials, then the set is called apolynomial polyhedron. Every analytic polyhedron is adomain of holomorphyand it is thuspseudo-convex.
The boundary of an analytic polyhedron is contained in the union of the set of hypersurfaces
An analytic polyhedron is aWeil polyhedron, orWeil domainif the intersection of anykof the above hypersurfaces has dimension no greater than2n-k.[2]
Thismathematical analysis–related article is astub. You can help Wikipedia byexpanding it.
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Inmathematics, aCaccioppoli setis a subset ofRn{\displaystyle \mathbb {R} ^{n}}whoseboundaryis (in a suitable sense)measurableand has (at leastlocally) afinitemeasure. A synonym isset of (locally) finite perimeter. Basically, a set is a Caccioppoli set if itscharacteristic functionis afunction of bounded variation, and its perimeter is the total variation of the characteristic function.
The basic concept of a Caccioppoli set was first introduced by the Italian mathematicianRenato Caccioppoliin the paper (Caccioppoli 1927): considering a plane set or asurfacedefined on anopen setin theplane, he defined theirmeasureorareaas thetotal variationin the sense ofTonelliof their definingfunctions, i.e. of theirparametric equations,provided this quantity wasbounded. Themeasure of theboundary of a setwas defined as afunctional, precisely aset function, for the first time: also, being defined onopen sets, it can be defined on allBorel setsand its value can be approximated by the values it takes on an increasingnetofsubsets. Another clearly stated (and demonstrated) property of this functional was itslower semi-continuity.
In the paper (Caccioppoli 1928), he precised by using atriangular meshas an increasingnetapproximating the open domain, definingpositive and negative variationswhose sum is the total variation, i.e. thearea functional. His inspiring point of view, as he explicitly admitted, was those ofGiuseppe Peano, as expressed by thePeano-Jordan Measure:to associate to every portion of a surface anorientedplane area in a similar way as anapproximating chordis associated to a curve. Also, another theme found in this theory was theextension of afunctionalfrom asubspaceto the wholeambient space: the use of theorems generalizing theHahn–Banach theoremis frequently encountered in Caccioppoli research. However, the restricted meaning oftotal variationin the sense ofTonelliadded much complication to the formal development of the theory, and the use of a parametric description of the sets restricted its scope.
Lamberto Cesariintroduced the "right" generalization offunctions of bounded variationto the case of several variables only in 1936:[1]perhaps, this was one of the reasons that induced Caccioppoli to present an improved version of his theory only nearly 24 years later, in the talk (Caccioppoli 1953) at the IVUMICongress in October 1951, followed by five notes published in theRendicontiof theAccademia Nazionale dei Lincei. These notes were sharply criticized byLaurence Chisholm Youngin theMathematical Reviews.[2]
In 1952Ennio De Giorgipresented his first results, developing the ideas of Caccioppoli, on the definition of the measure of boundaries of sets at theSalzburgCongress of the Austrian Mathematical Society: he obtained this results by using a smoothing operator, analogous to amollifier, constructed from theGaussian function, independently proving some results of Caccioppoli. Probably he was led to study this theory by his teacher and friendMauro Picone, who had also been the teacher of Caccioppoli and was likewise his friend. De Giorgi met Caccioppoli in 1953 for the first time: during their meeting, Caccioppoli expressed a profound appreciation of his work, starting their lifelong friendship.[3]The same year he published his first paper on the topic i.e. (De Giorgi 1953): however, this paper and the closely following one did not attracted much interest from the mathematical community. It was only with the paper (De Giorgi 1954), reviewed again by Laurence Chisholm Young in the Mathematical Reviews,[4]that his approach to sets of finite perimeter became widely known and appreciated: also, in the review, Young revised his previous criticism on the work of Caccioppoli.
The last paper of De Giorgi on the theory ofperimeterswas published in 1958: in 1959, after the death of Caccioppoli, he started to call sets of finite perimeter "Caccioppoli sets". Two years laterHerbert FedererandWendell Flemingpublished their paper (Federer & Fleming 1960), changing the approach to the theory. Basically they introduced two new kind ofcurrents, respectivelynormal currentsandintegral currents: in a subsequent series of papers and in his famous treatise,[5]Federer showed that Caccioppoli sets are normalcurrentsof dimensionn{\displaystyle n}inn{\displaystyle n}-dimensionaleuclidean spaces. However, even if the theory of Caccioppoli sets can be studied within the framework of theory ofcurrents, it is customary to study it through the "traditional" approach usingfunctions of bounded variation, as the various sections found in a lot of importantmonographsinmathematicsandmathematical physicstestify.[6]
In what follows, the definition and properties offunctions of bounded variationin then{\displaystyle n}-dimensional setting will be used.
Definition 1. LetΩ{\displaystyle \Omega }be anopen subsetofRn{\displaystyle \mathbb {R} ^{n}}and letE{\displaystyle E}be aBorel set. TheperimeterofE{\displaystyle E}inΩ{\displaystyle \Omega }is defined as follows
whereχE{\displaystyle \chi _{E}}is thecharacteristic functionofE{\displaystyle E}. That is, the perimeter ofE{\displaystyle E}in an open setΩ{\displaystyle \Omega }is defined to be thetotal variationof itscharacteristic functionon that open set. IfΩ=Rn{\displaystyle \Omega =\mathbb {R} ^{n}}, then we writeP(E)=P(E,Rn){\displaystyle P(E)=P(E,\mathbb {R} ^{n})}for the (global) perimeter.
Definition 2. TheBorel setE{\displaystyle E}is aCaccioppoli setif and only if it has finite perimeter in everyboundedopen subsetΩ{\displaystyle \Omega }ofRn{\displaystyle \mathbb {R} ^{n}}, i.e.
Therefore, a Caccioppoli set has acharacteristic functionwhosetotal variationis locally bounded. From the theory offunctions of bounded variationit is known that this implies the existence of avector-valuedRadon measureDχE{\displaystyle D\chi _{E}}such that
As noted for the case of generalfunctions of bounded variation, this vectormeasureDχE{\displaystyle D\chi _{E}}is thedistributionalorweakgradientofχE{\displaystyle \chi _{E}}. The total variation measure associated withDχE{\displaystyle D\chi _{E}}is denoted by|DχE|{\displaystyle |D\chi _{E}|}, i.e. for every open setΩ⊂Rn{\displaystyle \Omega \subset \mathbb {R} ^{n}}we write|DχE|(Ω){\displaystyle |D\chi _{E}|(\Omega )}forP(E,Ω)=V(χE,Ω){\displaystyle P(E,\Omega )=V(\chi _{E},\Omega )}.
In his papers (De Giorgi 1953) and (De Giorgi 1954),Ennio De Giorgiintroduces the followingsmoothingoperator, analogous to theWeierstrass transformin the one-dimensionalcase
As one can easily prove,Wλχ(x){\displaystyle W_{\lambda }\chi (x)}is asmooth functionfor allx∈Rn{\displaystyle x\in \mathbb {R} ^{n}}, such that
also, itsgradientis everywhere well defined, and so is itsabsolute value
Having defined this function, De Giorgi gives the following definition ofperimeter:
Definition 3. LetΩ{\displaystyle \Omega }be anopen subsetofRn{\displaystyle \mathbb {R} ^{n}}and letE{\displaystyle E}be aBorel set. TheperimeterofE{\displaystyle E}inΩ{\displaystyle \Omega }is the value
Actually De Giorgi considered the caseΩ=Rn{\displaystyle \Omega =\mathbb {R} ^{n}}: however, the extension to the general case is not difficult. It can be proved that the two definitions are exactly equivalent: for a proof see the already cited De Giorgi's papers or the book (Giusti 1984). Now having defined what a perimeter is, De Giorgi gives the same definition 2 of what a set of(locally) finiteperimeter is.
The following properties are the ordinary properties which the general notion of aperimeteris supposed to have:
For any given Caccioppoli setE⊂Rn{\displaystyle E\subset \mathbb {R} ^{n}}there exist two naturally associated analytic quantities: the vector-valuedRadon measureDχE{\displaystyle D\chi _{E}}and itstotal variation measure|DχE|{\displaystyle |D\chi _{E}|}. Given that
is the perimeter within any open setΩ{\displaystyle \Omega }, one should expect thatDχE{\displaystyle D\chi _{E}}alone should somehow account for the perimeter ofE{\displaystyle E}.
It is natural to try to understand the relationship between the objectsDχE{\displaystyle D\chi _{E}},|DχE|{\displaystyle |D\chi _{E}|}, and thetopological boundary∂E{\displaystyle \partial E}. There is an elementary lemma that guarantees that thesupport(in the sense ofdistributions) ofDχE{\displaystyle D\chi _{E}}, and therefore also|DχE|{\displaystyle |D\chi _{E}|}, is alwayscontainedin∂E{\displaystyle \partial E}:
Lemma. The support of the vector-valued Radon measureDχE{\displaystyle D\chi _{E}}is asubsetof thetopological boundary∂E{\displaystyle \partial E}ofE{\displaystyle E}.
Proof. To see this choosex0∉∂E{\displaystyle x_{0}\notin \partial E}: thenx0{\displaystyle x_{0}}belongs to theopen setRn∖∂E{\displaystyle \mathbb {R} ^{n}\setminus \partial E}and this implies that it belongs to anopen neighborhoodA{\displaystyle A}contained in theinteriorofE{\displaystyle E}or in the interior ofRn∖E{\displaystyle \mathbb {R} ^{n}\setminus E}. Letϕ∈Cc1(A;Rn){\displaystyle \phi \in C_{c}^{1}(A;\mathbb {R} ^{n})}. IfA⊆(Rn∖E)∘=Rn∖E−{\displaystyle A\subseteq (\mathbb {R} ^{n}\setminus E)^{\circ }=\mathbb {R} ^{n}\setminus E^{-}}whereE−{\displaystyle E^{-}}is theclosureofE{\displaystyle E}, thenχE(x)=0{\displaystyle \chi _{E}(x)=0}forx∈A{\displaystyle x\in A}and
Likewise, ifA⊆E∘{\displaystyle A\subseteq E^{\circ }}thenχE(x)=1{\displaystyle \chi _{E}(x)=1}forx∈A{\displaystyle x\in A}so
Withϕ∈Cc1(A,Rn){\displaystyle \phi \in C_{c}^{1}(A,\mathbb {R} ^{n})}arbitrary it follows thatx0{\displaystyle x_{0}}is outside the support ofDχE{\displaystyle D\chi _{E}}.
The topological boundary∂E{\displaystyle \partial E}turns out to be too crude for Caccioppoli sets because itsHausdorff measureovercompensates for the perimeterP(E){\displaystyle P(E)}defined above. Indeed, the Caccioppoli set
representing a square together with a line segment sticking out on the left has perimeterP(E)=4{\displaystyle P(E)=4}, i.e. the extraneous line segment is ignored, while its topological boundary
has one-dimensional Hausdorff measureH1(∂E)=5{\displaystyle {\mathcal {H}}^{1}(\partial E)=5}.
The "correct" boundary should therefore be a subset of∂E{\displaystyle \partial E}. We define:
Definition 4. Thereduced boundaryof a Caccioppoli setE⊂Rn{\displaystyle E\subset \mathbb {R} ^{n}}is denoted by∂∗E{\displaystyle \partial ^{*}E}and is defined to be equal to be the collection of pointsx{\displaystyle x}at which the limit:
exists and has length equal to one, i.e.|νE(x)|=1{\displaystyle |\nu _{E}(x)|=1}.
One can remark that by theRadon-Nikodym Theoremthe reduced boundary∂∗E{\displaystyle \partial ^{*}E}is necessarily contained in the support ofDχE{\displaystyle D\chi _{E}}, which in turn is contained in the topological boundary∂E{\displaystyle \partial E}as explained in the section above. That is:
The inclusions above are not necessarily equalities as the previous example shows. In that example,∂E{\displaystyle \partial E}is the square with the segment sticking out,supportDχE{\displaystyle \operatorname {support} D\chi _{E}}is the square, and∂∗E{\displaystyle \partial ^{*}E}is the square without its four corners.
For convenience, in this section we treat only the case whereΩ=Rn{\displaystyle \Omega =\mathbb {R} ^{n}}, i.e. the setE{\displaystyle E}has (globally) finite perimeter. De Giorgi's theorem provides geometric intuition for the notion of reduced boundaries and confirms that it is the more natural definition for Caccioppoli sets by showing
i.e. that itsHausdorff measureequals the perimeter of the set. The statement of the theorem is quite long because it interrelates various geometric notions in one fell swoop.
Theorem. SupposeE⊂Rn{\displaystyle E\subset \mathbb {R} ^{n}}is a Caccioppoli set. Then at each pointx{\displaystyle x}of the reduced boundary∂∗E{\displaystyle \partial ^{*}E}there exists a multiplicity oneapproximate tangent spaceTx{\displaystyle T_{x}}of|DχE|{\displaystyle |D\chi _{E}|}, i.e. a codimension-1 subspaceTx{\displaystyle T_{x}}ofRn{\displaystyle \mathbb {R} ^{n}}such that
for every continuous, compactly supportedf:Rn→R{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }. In fact the subspaceTx{\displaystyle T_{x}}is theorthogonal complementof the unit vector
defined previously. This unit vector also satisfies
locally inL1{\displaystyle L^{1}}, so it is interpreted as an approximate inward pointingunitnormal vectorto the reduced boundary∂∗E{\displaystyle \partial ^{*}E}. Finally,∂∗E{\displaystyle \partial ^{*}E}is (n-1)-rectifiableand the restriction of (n-1)-dimensionalHausdorff measureHn−1{\displaystyle {\mathcal {H}}^{n-1}}to∂∗E{\displaystyle \partial ^{*}E}is|DχE|{\displaystyle |D\chi _{E}|}, i.e.
In other words, up toHn−1{\displaystyle {\mathcal {H}}^{n-1}}-measure zero the reduced boundary∂∗E{\displaystyle \partial ^{*}E}is the smallest set on whichDχE{\displaystyle D\chi _{E}}is supported.
From the definition of the vectorRadon measureDχE{\displaystyle D\chi _{E}}and from the properties of the perimeter, the following formula holds true:
This is one version of thedivergence theoremfordomainswith non smoothboundary. De Giorgi's theorem can be used to formulate the same identity in terms of the reduced boundary∂∗E{\displaystyle \partial ^{*}E}and the approximate inward pointing unit normal vectorνE{\displaystyle \nu _{E}}. Precisely, the following equality holds
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Inmathematics, aHermitian symmetric spaceis aHermitian manifoldwhich at every point has an inversion symmetry preserving the Hermitian structure. First studied byÉlie Cartan, they form a natural generalization of the notion ofRiemannian symmetric spacefromreal manifoldstocomplex manifolds.
Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, asBorelshowed, can be embedded as an open subspace of its compact dual space.Harish Chandrashowed that each non-compact space can be realized as abounded symmetric domainin a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is theunit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex planeC. The one-point compactification ofC, theRiemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C).
Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain a maximal torus and have center isomorphic to the circle group. There is a complete classification of irreducible spaces, with four classical series, studied by Cartan, and two exceptional cases; the classification can be deduced fromBorel–de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus. Hermitian symmetric spaces appear in the theory ofJordan triple systems,several complex variables,complex geometry,automorphic formsandgroup representations, in particular permitting the construction of theholomorphic discrete series representationsof semisimple Lie groups.[1]
LetHbe a connected compact semisimple Lie group, σ an automorphism ofHof order 2 andHσthe fixed point subgroup of σ. LetKbe a closed subgroup ofHlying betweenHσand itsidentity component. The compact homogeneous spaceH/Kis called asymmetric space of compact type. The Lie algebrah{\displaystyle {\mathfrak {h}}}admits a decomposition
wherek{\displaystyle {\mathfrak {k}}}, the Lie algebra ofK, is the +1 eigenspace of σ andm{\displaystyle {\mathfrak {m}}}the –1 eigenspace. Ifk{\displaystyle {\mathfrak {k}}}contains no simple summand ofh{\displaystyle {\mathfrak {h}}}, the pair (h{\displaystyle {\mathfrak {h}}}, σ) is called anorthogonal symmetric Lie algebraofcompact type.[2]
Any inner product onh{\displaystyle {\mathfrak {h}}}, invariant under theadjoint representationand σ, induces a Riemannian structure onH/K, withHacting by isometries. A canonical example is given by minus theKilling form. Under such an inner product,k{\displaystyle {\mathfrak {k}}}andm{\displaystyle {\mathfrak {m}}}are orthogonal.H/Kis then a Riemannian symmetric space of compact type.[3]
The symmetric spaceH/Kis called aHermitian symmetric spaceif it has analmost complex structurepreserving the Riemannian metric. This is equivalent to the existence of a linear mapJwithJ2= −Ionm{\displaystyle {\mathfrak {m}}}which preserves the inner product and commutes with the action ofK.
If (h{\displaystyle {\mathfrak {h}}},σ) is Hermitian,Khas non-trivial center and the symmetry σ is inner, implemented by an element of the center ofK.
In factJlies ink{\displaystyle {\mathfrak {k}}}and exptJforms a one-parameter group in the center ofK. This follows because ifA,B,C,Dlie inm{\displaystyle {\mathfrak {m}}}, then by the invariance of the inner product onh{\displaystyle {\mathfrak {h}}}[4]
ReplacingAandBbyJAandJB, it follows that
Define a linear map δ onh{\displaystyle {\mathfrak {h}}}by extendingJto be 0 onk{\displaystyle {\mathfrak {k}}}. The last relation shows that δ is a derivation ofh{\displaystyle {\mathfrak {h}}}. Sinceh{\displaystyle {\mathfrak {h}}}is semisimple, δ must be an inner derivation, so that
withTink{\displaystyle {\mathfrak {k}}}andAinm{\displaystyle {\mathfrak {m}}}. TakingXink{\displaystyle {\mathfrak {k}}}, it follows thatA= 0 andTlies in the center ofk{\displaystyle {\mathfrak {k}}}and hence thatKis non-semisimple. The symmetry σ is implemented byz= exp πTand the almost complex structure by exp π/2T.[5]
The innerness of σ implies thatKcontains a maximal torus ofH, so has maximal rank. On the other hand, the centralizer of the subgroup generated by the torusSof elements exptTis connected, since ifxis any element inKthere is a maximal torus containingxandS, which lies in the centralizer. On the other hand, it containsKsinceSis central inKand is contained inKsincezlies inS. SoKis the centralizer ofSand hence connected. In particularKcontains the center ofH.[2]
The symmetric space or the pair (h{\displaystyle {\mathfrak {h}}}, σ) is said to beirreducibleif the adjoint action ofk{\displaystyle {\mathfrak {k}}}(or equivalently the identity component ofHσorK) is irreducible onm{\displaystyle {\mathfrak {m}}}. This is equivalent to the maximality ofk{\displaystyle {\mathfrak {k}}}as a subalgebra.[6]
In fact there is a one-one correspondence between intermediate subalgebrasl{\displaystyle {\mathfrak {l}}}andK-invariant subspacesm1{\displaystyle {\mathfrak {m}}_{1}}ofm{\displaystyle {\mathfrak {m}}}given by
Any orthogonal symmetric algebra (g{\displaystyle {\mathfrak {g}}}, σ) of Hermitian type can be decomposed as an (orthogonal) direct sum of irreducible orthogonal symmetric algebras of Hermitian type.[7]
In facth{\displaystyle {\mathfrak {h}}}can be written as a direct sum of simple algebras
each of which is left invariant by the automorphism σ and the complex structureJ, since they are both inner. The eigenspace decomposition ofh1{\displaystyle {\mathfrak {h}}_{1}}coincides with its intersections withk{\displaystyle {\mathfrak {k}}}andm{\displaystyle {\mathfrak {m}}}. So the restriction of σ toh1{\displaystyle {\mathfrak {h}}_{1}}is irreducible.
This decomposition of the orthogonal symmetric Lie algebra yields a direct product decomposition of the corresponding compact symmetric spaceH/KwhenHis simply connected. In this case the fixed point subgroupHσis automatically connected. For simply connectedH, the symmetric spaceH/Kis the direct product ofHi/KiwithHisimply connected and simple. In the irreducible case,Kis a maximal connected subgroup ofH. SinceKacts irreducibly onm{\displaystyle {\mathfrak {m}}}(regarded as a complex space for the complex structure defined byJ), the center ofKis a one-dimensional torusT, given by the operators exptT. Since eachHis simply connected andKconnected, the quotientH/Kis simply connected.[8]
ifH/Kis irreducible withKnon-semisimple, the compact groupHmust be simple andKof maximal rank. FromBorel-de Siebenthal theory, the involution σ is inner andKis the centralizer of its center, which is isomorphic toT. In particularKis connected. It follows thatH/Kis simply connected and there is aparabolic subgroupPin thecomplexificationGofHsuch thatH/K=G/P. In particular there is a complex structure onH/Kand the action ofHis holomorphic. Since any Hermitian symmetric space is a product of irreducible spaces, the same is true in general.
At theLie algebralevel, there is a symmetric decomposition
where(m,J){\displaystyle ({\mathfrak {m}},J)}is a real vector space with a complex structureJ, whose complex dimension is given in the table. Correspondingly, there is agraded Lie algebradecomposition
wherem⊗C=m−⊕m+{\displaystyle {\mathfrak {m}}\otimes \mathbb {C} ={\mathfrak {m}}_{-}\oplus {\mathfrak {m}}_{+}}is the decomposition into +iand −ieigenspaces ofJandl=k⊗C{\displaystyle {\mathfrak {l}}={\mathfrak {k}}\otimes \mathbb {C} }. The Lie algebra ofPis the semidirect productm+⊕l{\displaystyle {\mathfrak {m}}^{+}\oplus {\mathfrak {l}}}. The complex Lie algebrasm±{\displaystyle {\mathfrak {m}}_{\pm }}are Abelian. Indeed, ifUandVlie inm±{\displaystyle {\mathfrak {m}}_{\pm }}, [U,V] =J[U,V] = [JU,JV] = [±iU,±iV] = –[U,V], so the Lie bracket must vanish.
The complex subspacesm±{\displaystyle {\mathfrak {m}}_{\pm }}ofmC{\displaystyle {\mathfrak {m}}_{\mathbb {C} }}are irreducible for the action ofK, sinceJcommutes withKso that each is isomorphic tom{\displaystyle {\mathfrak {m}}}with complex structure ±J. Equivalently the centreTofKacts onm+{\displaystyle {\mathfrak {m}}_{+}}by the identity representation and onm−{\displaystyle {\mathfrak {m}}_{-}}by its conjugate.[9]
The realization ofH/Kas a generalized flag varietyG/Pis obtained by takingGas in the table (thecomplexificationofH) andPto be the parabolic subgroup equal to the semidirect product ofL, the complexification ofK, with the complex Abelian subgroup expm+{\displaystyle {\mathfrak {m}}_{+}}. (In the language ofalgebraic groups,Lis theLevi factorofP.)
Any Hermitian symmetric space of compact type is simply connected and can be written as a direct product of irreducible hermitian symmetric spacesHi/KiwithHisimple,Kiconnected of maximal rank with centerT. The irreducible ones are therefore exactly the non-semisimple cases classified byBorel–de Siebenthal theory.[2]
Accordingly, the irreducible compact Hermitian symmetric spacesH/Kare classified as follows.
In terms of the classification of compact Riemannian symmetric spaces, the Hermitian symmetric spaces are the four infinite series AIII, DIII, CI and BDI withp= 2 orq= 2, and two exceptional spaces, namely EIII and EVII.
The irreducible Hermitian symmetric spaces of compact type are all simply connected. The corresponding symmetry σ of the simply connected simple compact Lie group is inner, given by conjugation by the unique elementSinZ(K) /Z(H) of period 2. For the classical groups, as in the table above, these symmetries are as follows:[10]
The maximal parabolic subgroupPcan be described explicitly in these classical cases. For AIII
in SL(p+q,C).P(p,q) is the stabilizer of a subspace of dimensionpinCp+q.
The other groups arise as fixed points of involutions. LetJbe then×nmatrix with 1's on the antidiagonal and 0's elsewhere and set
Then Sp(n,C) is the fixed point subgroup of the involution θ(g) =A(gt)−1A−1of SL(2n,C). SO(n,C) can be realised as the fixed points of ψ(g) =B(gt)−1B−1in SL(n,C) whereB=J. These involutions leave invariantP(n,n) in the cases DIII and CI andP(p,2) in the case BDI. The corresponding parabolic subgroupsPare obtained by taking the fixed points. The compact groupHacts transitively onG/P, so thatG/P=H/K.
As with symmetric spaces in general, each compact Hermitian symmetric spaceH/Khas a noncompact dualH*/Kobtained by replacingHwith the closed real Lie subgroupH*of the complex Lie groupGwith Lie algebra
Whereas the natural map fromH/KtoG/Pis an isomorphism, the natural map fromH*/KtoG/Pis only an inclusion onto an open subset. This inclusion is called theBorel embeddingafterArmand Borel. In factP∩H=K=P∩H*. The images ofHandH* have the same dimension so are open. Since the image ofHis compact, so closed, it follows thatH/K=G/P.[11]
The polar decomposition in the complex linear groupGimplies the Cartan decompositionH* =K⋅ expim{\displaystyle i{\mathfrak {m}}}inH*.[12]
Moreover, given a maximal Abelian subalgebraa{\displaystyle {\mathfrak {a}}}in t,A= expa{\displaystyle {\mathfrak {a}}}is a toral subgroup such that σ(a) =a−1onA; and any two sucha{\displaystyle {\mathfrak {a}}}'s are conjugate by an element ofK. A similar statement holds fora∗=ia{\displaystyle {\mathfrak {a}}^{*}=i{\mathfrak {a}}}. Morevoer ifA* = expa∗{\displaystyle {\mathfrak {a}}^{*}}, then
These results are special cases of the Cartan decomposition in any Riemannian symmetric space and its dual. The geodesics emanating from the origin in the homogeneous spaces can be identified with one parameter groups with generators inim{\displaystyle i{\mathfrak {m}}}orm{\displaystyle {\mathfrak {m}}}. Similar results hold for in the compact case:H=K⋅ expim{\displaystyle i{\mathfrak {m}}}andH=KAK.[8]
The properties of thetotally geodesicsubspaceAcan be shown directly.Ais closed because the closure ofAis a toral subgroup satisfying σ(a) =a−1, so its Lie algebra lies inm{\displaystyle {\mathfrak {m}}}and hence equalsa{\displaystyle {\mathfrak {a}}}by maximality.Acan be generated topologically by a single element expX, soa{\displaystyle {\mathfrak {a}}}is the centralizer ofXinm{\displaystyle {\mathfrak {m}}}. In theK-orbit of any element ofm{\displaystyle {\mathfrak {m}}}there is an elementYsuch that (X,AdkY) is minimized atk= 1. Settingk= exptTwithTink{\displaystyle {\mathfrak {k}}}, it follows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so thatYmust lie ina{\displaystyle {\mathfrak {a}}}. Thusm{\displaystyle {\mathfrak {m}}}is the union of the conjugates ofa{\displaystyle {\mathfrak {a}}}. In particular some conjugate ofXlies in any other choice ofa{\displaystyle {\mathfrak {a}}}, which centralizes that conjugate; so by maximality the only possibilities are conjugates ofa{\displaystyle {\mathfrak {a}}}.[13]
The decompositions
can be proved directly by applying theslice theoremforcompact transformation groupsto the action ofKonH/K.[14]In fact the spaceH/Kcan be identified with
a closed submanifold ofH, and the Cartan decomposition follows by showing thatMis the union of thekAk−1forkinK. Since this union is the continuous image ofK×A, it is compact and connected. So it suffices to show that the union is open inMand for this it is enough to show eachainAhas an open neighbourhood in this union. Now by computing derivatives at 0, the union contains an open neighbourhood of 1. Ifais central the union is invariant under multiplication bya, so contains an open neighbourhood ofa. Ifais not central, writea=b2withbinA. Then τ = Adb− Adb−1is a skew-adjoint operator onh{\displaystyle {\mathfrak {h}}}anticommuting with σ, which can be regarded as aZ2-grading operator σ onh{\displaystyle {\mathfrak {h}}}. By anEuler–Poincaré characteristicargument it follows that the superdimension ofh{\displaystyle {\mathfrak {h}}}coincides with the superdimension of the kernel of τ. In other words,
whereka{\displaystyle {\mathfrak {k}}_{a}}andma{\displaystyle {\mathfrak {m}}_{a}}are the subspaces fixed by Ada. Let the orthogonal complement ofka{\displaystyle {\mathfrak {k}}_{a}}ink{\displaystyle {\mathfrak {k}}}beka⊥{\displaystyle {\mathfrak {k}}_{a}^{\perp }}. Computing derivatives, it follows that AdeX(aeY), whereXlies inka⊥{\displaystyle {\mathfrak {k}}_{a}^{\perp }}andYinma{\displaystyle {\mathfrak {m}}_{a}}, is an open neighbourhood ofain the union. Here the termsaeYlie in the union by the argument for centrala: indeedais in the center of the identity component of the centralizer ofawhich is invariant under σ and containsA.
The dimension ofa{\displaystyle {\mathfrak {a}}}is called therankof the Hermitian symmetric space.
In the case of Hermitian symmetric spaces, Harish-Chandra gave a canonical choice fora{\displaystyle {\mathfrak {a}}}.
This choice ofa{\displaystyle {\mathfrak {a}}}is determined by taking a maximal torusTofHinKwith Lie algebrat{\displaystyle {\mathfrak {t}}}. Since the symmetry σ is implemented by an element ofTlying in the centre ofH, the root spacesgα{\displaystyle {\mathfrak {g}}_{\alpha }}ing{\displaystyle {\mathfrak {g}}}are left invariant by σ. It acts as the identity on those contained inkC{\displaystyle {\mathfrak {k}}_{\mathbb {C} }}and minus the identity on those inmC{\displaystyle {\mathfrak {m}}_{\mathbb {C} }}.
The roots with root spaces inkC{\displaystyle {\mathfrak {k}}_{\mathbb {C} }}are calledcompact rootsand those with root spaces inmC{\displaystyle {\mathfrak {m}}_{\mathbb {C} }}are callednoncompact roots. (This terminology originates from the symmetric space of noncompact type.) IfHis simple, the generatorZof the centre ofKcan be used to define a set of positive roots, according to the sign of α(Z). With this choice of rootsm+{\displaystyle {\mathfrak {m}}_{+}}andm−{\displaystyle {\mathfrak {m}}_{-}}are the direct sum of the root spacesgα{\displaystyle {\mathfrak {g}}_{\alpha }}over positive and negative noncompact roots α. Root vectorsEαcan be chosen so that
lie inh{\displaystyle {\mathfrak {h}}}. The simple roots α1, ...., αnare the indecomposable positive roots. These can be numbered so that αivanishes on the center ofh{\displaystyle {\mathfrak {h}}}fori, whereas α1does not. Thus α1is the unique noncompact simple root and the other simple roots are compact. Any positive noncompact root then has the form β = α1+c2α2+ ⋅⋅⋅ +cnαnwith non-negative coefficientsci. These coefficients lead to alexicographic orderon positive roots. The coefficient of α1is always one becausem−{\displaystyle {\mathfrak {m}}_{-}}is irreducible forKso is spanned by vectors obtained by successively applying the lowering operatorsE–αfor simple compact roots α.
Two roots α and β are said to bestrongly orthogonalif ±α ±β are not roots or zero, written α ≐ β. The highest positive root ψ1is noncompact. Take ψ2to be the highest noncompact positive root strongly orthogonal to ψ1(for the lexicographic order). Then continue in this way taking ψi+ 1to be the highest noncompact positive root strongly orthogonal to ψ1, ..., ψiuntil the process terminates. The corresponding vectors
lie inm{\displaystyle {\mathfrak {m}}}and commute by strong orthogonality. Their spana{\displaystyle {\mathfrak {a}}}is Harish-Chandra's canonical maximal Abelian subalgebra.[15](As Sugiura later showed, having fixedT, the set of strongly orthogonal roots is uniquely determined up to applying an element in the Weyl group ofK.[16])
Maximality can be checked by showing that if
for alli, thencα= 0 for all positive noncompact roots α different from the ψj's. This follows by showing inductively that ifcα≠ 0, then α is strongly orthogonal to ψ1, ψ2, ... a contradiction. Indeed, the above relation shows ψi+ α cannot be a root; and that if ψi– α is a root, then it would necessarily have the form β – ψi. If ψi– α were negative, then α would be a higher positive root than ψi, strongly orthogonal to the ψjwithj<i, which is not possible; similarly if β – ψiwere positive.
Harish-Chandra's canonical choice ofa{\displaystyle {\mathfrak {a}}}leads to a polydisk and polysphere theorem inH*/KandH/K. This result reduces the geometry to products of the prototypic example involving SL(2,C), SU(1,1) and SU(2), namely the unit disk inside the Riemann sphere.
In the case ofH= SU(2) the symmetry σ is given by conjugation by the diagonal matrix with entries ±iso that
The fixed point subgroup is the maximal torusT, the diagonal matrices with entriese±it. SU(2) acts on the Riemann sphereCP1{\displaystyle \mathbf {CP} ^{1}}transitively by Möbius transformations andTis the stabilizer of 0. SL(2,C), the complexification of SU(2), also acts by Möbius transformations and the stabiliser of 0 is the subgroupBof lower triangular matrices. The noncompact subgroup SU(1,1) acts with precisely three orbits: the open unit disk |z| < 1; the unit circlez= 1; and its exterior |z| > 1. Thus
whereB+andTCdenote the subgroups of upper triangular and diagonal matrices in SL(2,C). The middle term is the orbit of 0 under the upper unitriangular matrices
Now for each root ψithere is a homomorphism of πiof SU(2) intoHwhich is compatible with the symmetries. It extends uniquely to a homomorphism of SL(2,C) intoG. The images of the Lie algebras for different ψi's commute since they are strongly orthogonal. Thus there is a homomorphism π of the direct product SU(2)rintoHcompatible with the symmetries. It extends to a homomorphism of SL(2,C)rintoG. The kernel of π is contained in the center (±1)rof SU(2)rwhich is fixed pointwise by the symmetry. So the image of the center under π lies inK. Thus there is an embedding of the polysphere (SU(2)/T)rintoH/K=G/Pand the polysphere contains the polydisk (SU(1,1)/T)r. The polysphere and polydisk are the direct product ofrcopies of the Riemann sphere and the unit disk. By the Cartan decompositions in SU(2) and SU(1,1),
the polysphere is the orbit ofTrAinH/Kand the polydisk is the orbit ofTrA*, whereTr= π(Tr) ⊆K. On the other hand,H=KAKandH* =KA*K.
Hence every element in the compact Hermitian symmetric spaceH/Kis in theK-orbit of a point in the polysphere; and every element in the image under the Borel embedding of the noncompact Hermitian symmetric spaceH* /Kis in theK-orbit of a point in the polydisk.[17]
H* /K, the Hermitian symmetric space of noncompact type, lies in the image ofexpm+{\displaystyle \exp {\mathfrak {m}}_{+}}, a dense open subset ofH/Kbiholomorphic tom+{\displaystyle {\mathfrak {m}}_{+}}. The corresponding domain inm+{\displaystyle {\mathfrak {m}}_{+}}is bounded. This is theHarish-Chandra embeddingnamed afterHarish-Chandra.
In fact Harish-Chandra showed the following properties of the spaceX=exp(m+)⋅KC⋅exp(m−)=exp(m+)⋅P{\displaystyle \mathbf {X} =\exp({\mathfrak {m}}_{+})\cdot K_{\mathbb {C} }\cdot \exp({\mathfrak {m}}_{-})=\exp({\mathfrak {m}}_{+})\cdot P}:
In factM±=expm±{\displaystyle M_{\pm }=\exp {\mathfrak {m}}_{\pm }}are complex Abelian groups normalised byKC. Moreover,[m+,m−]⊂kC{\displaystyle [{\mathfrak {m}}_{+},{\mathfrak {m}}_{-}]\subset {\mathfrak {k}}_{\mathfrak {C}}}since[m,m]⊂k{\displaystyle [{\mathfrak {m}},{\mathfrak {m}}]\subset {\mathfrak {k}}}.
This impliesP∩M+= {1}. For ifx=eXwithXinm+{\displaystyle {\mathfrak {m}}_{+}}lies inP, it must normalizeM−and hencem−{\displaystyle {\mathfrak {m}}_{-}}. But ifYlies inm−{\displaystyle {\mathfrak {m}}_{-}}, then
so thatXcommutes withm−{\displaystyle {\mathfrak {m}}_{-}}. But ifXcommutes with every noncompact root space, it must be 0, sox= 1. It follows that the multiplication map μ onM+×Pis injective so (1) follows. Similarly the derivative of μ at (x,p) is
which is injective, so (2) follows. For the special caseH= SU(2),H* = SU(1,1) andG= SL(2,C) the remaining assertions are consequences of the identification with the Riemann sphere,Cand unit disk. They can be applied to the groups defined for each root ψi. By the polysphere and polydisk theoremH*/K,X/PandH/Kare the union of theK-translates of the polydisk,Crand the polysphere. SoH* lies inX, the closure ofH*/Kis compact inX/P, which is in turn dense inH/K.
Note that (2) and (3) are also consequences of the fact that the image ofXinG/Pis that of the big cellB+Bin theGauss decompositionofG.[18]
Using results on therestricted root systemof the symmetric spacesH/KandH*/K,Hermannshowed that the image ofH*/Kinm+{\displaystyle {\mathfrak {m}}_{+}}is a generalized unit disk. In fact it is theconvex setofXfor which theoperator normof ad ImXis less than one.[19]
A bounded domainΩin a complex vector space is said to be abounded symmetric domainif for everyxinΩ, there is an involutive biholomorphismσxofΩfor whichxis an isolated fixed point. The Harish-Chandra embedding exhibits every Hermitian symmetric space of noncompact typeH* /Kas a bounded symmetric domain. The biholomorphism group ofH*/Kis equal to its isometry groupH*.
Conversely every bounded symmetric domain arises in this way. Indeed, given a bounded symmetric domainΩ, theBergman kerneldefines ametriconΩ, theBergman metric, for which every biholomorphism is an isometry. This realizesΩas a Hermitian symmetric space of noncompact type.[20]
The irreducible bounded symmetric domains are calledCartan domainsand are classified as follows.
In the classical cases (I–IV), the noncompact group can be realized by 2 × 2 block matrices[21]
acting by generalizedMöbius transformations
The polydisk theorem takes the following concrete form in the classical cases:[22]
The noncompact groupH* acts on the complex Hermitian symmetric spaceH/K=G/Pwith only finitely many orbits. The orbit structure is described in detail inWolf (1972). In particular the closure of the bounded domainH*/Khas a unique closed orbit, which is theShilov boundaryof the domain. In general the orbits are unions of Hermitian symmetric spaces of lower dimension. The complex function theory of the domains, in particular the analogue of theCauchy integral formulas, are described for the Cartan domains inHua (1979). The closure of the bounded domain is theBaily–Borel compactificationofH*/K.[23]
The boundary structure can be described usingCayley transforms. For each copy of SU(2) defined by one of the noncompact roots ψi, there is a Cayley transformciwhich as a Möbius transformation maps the unit disk onto the upper half plane. Given a subsetIof indices of the strongly orthogonal family ψ1, ..., ψr, thepartial Cayley transformcIis defined as the product of theci's withiinIin the product of the groups πi. LetG(I) be the centralizer of this product inGandH*(I) =H* ∩G(I). Since σ leavesH*(I) invariant, there is a corresponding Hermitian symmetric spaceMIH*(I)/H*(I)∩K⊂H*/K=M. The boundary component for the subsetIis the union of theK-translates ofcIMI. WhenIis the set of all indices,MIis a single point and the boundary component is the Shilov boundary. Moreover,MIis in the closure ofMJif and only ifI⊇J.[24]
Every Hermitian symmetric space is aKähler manifold. They can be defined equivalently as Riemannian symmetric spaces with a parallel complex structure with respect to which the Riemannian metric isHermitian. The complex structure is automatically preserved by the isometry groupHof the metric, and so any Hermitian symmetric spaceMis a homogeneous complex manifold. Some examples arecomplex vector spacesandcomplex projective spaces, with their usual Hermitian metrics andFubini–Study metrics, and the complexunit ballswith suitable metrics so that they becomecompleteand Riemannian symmetric. ThecompactHermitian symmetric spaces areprojective varieties, and admit a strictly largerLie groupGofbiholomorphismswith respect to which they are homogeneous: in fact, they aregeneralized flag manifolds, i.e.,Gissemisimpleand the stabilizer of a point is a parabolic subgroupPofG. Among (complex) generalized flag manifoldsG/P, they are characterized as those for which thenilradicalof the Lie algebra ofPis abelian. Thus they are contained within the family of symmetric R-spaces which conversely comprises Hermitian symmetric spaces and their real forms. The non-compact Hermitian symmetric spaces can be realized as bounded domains in complex vector spaces.
Although the classical Hermitian symmetric spaces can be constructed by ad hoc methods,Jordan triple systems, or equivalently Jordan pairs, provide a uniform algebraic means of describing all the basic properties connected with a Hermitian symmetric space of compact type and its non-compact dual. This theory is described in detail inKoecher (1969)andLoos (1977)and summarized inSatake (1981). The development is in the reverse order from that using the structure theory of compact Lie groups. It starting point is the Hermitian symmetric space of noncompact type realized as a bounded symmetric domain. It can be described in terms of aJordan pairor hermitianJordan triple system. This Jordan algebra structure can be used to reconstruct the dual Hermitian symmetric space of compact type, including in particular all the associated Lie algebras and Lie groups.
The theory is easiest to describe when the irreducible compact Hermitian symmetric space is of tube type. In that case the space is determined by a simple real Lie algebrag{\displaystyle {\mathfrak {g}}}with negative definite Killing form. It must admit an action of SU(2) which only acts via the trivial and adjoint representation, both types occurring. Sinceg{\displaystyle {\mathfrak {g}}}is simple, this action is inner, so implemented by an inclusion of the Lie algebra of SU(2) ing{\displaystyle {\mathfrak {g}}}. The complexification ofg{\displaystyle {\mathfrak {g}}}decomposes as a direct sum of three eigenspaces for the diagonal matrices in SU(2). It is a three-graded complex Lie algebra, with the Weyl group element of SU(2) providing the involution. Each of the ±1 eigenspaces has the structure of a unital complex Jordan algebra explicitly arising as the complexification of a Euclidean Jordan algebra. It can be identified with the multiplicity space of the adjoint representation of SU(2) ing{\displaystyle {\mathfrak {g}}}.
The description of irreducible Hermitian symmetric spaces of tube type starts from a simple Euclidean Jordan algebraE. It admitsJordan frames, i.e. sets of orthogonal minimal idempotentse1, ...,em. Any two are related by an automorphism ofE, so that the integermis an invariant called therankofE. Moreover, ifAis the complexification ofE, it has a unitarystructure group. It is a subgroup of GL(A) preserving the natural complex inner product onA. Any elementainAhas a polar decompositiona=uΣ αiaiwithαi≥ 0. The spectral norm is defined by ||a|| = sup αi. The associatedbounded symmetric domainis just the open unit ballDinA. There is a biholomorphism betweenDand the tube domainT=E+iCwhereCis the open self-dual convex cone of elements inEof the forma=uΣ αiaiwithuan automorphism ofEand αi> 0. This gives two descriptions of the Hermitian symmetric space of noncompact type. There is a natural way of usingmutationsof the Jordan algebraAto compactify the spaceA. The compactificationXis a complex manifold and the finite-dimensional Lie algebrag{\displaystyle {\mathfrak {g}}}of holomorphic vector fields onXcan be determined explicitly. One parameter groups of biholomorphisms can be defined such that the corresponding holomorphic vector fields spang{\displaystyle {\mathfrak {g}}}. This includes the group of all complex Möbius transformations corresponding to matrices in SL(2,C). The subgroup SU(1,1) leaves invariant the unit ball and its closure. The subgroup SL(2,R) leaves invariant the tube domain and its closure. The usual Cayley transform and its inverse, mapping the unit disk inCto the upper half plane, establishes analogous maps betweenDandT. The polydisk corresponds to the real and complex Jordan subalgebras generated by a fixed Jordan frame. It admits a transitive action of SU(2)mand this action extends toX. The groupGgenerated by the one-parameter groups of biholomorphisms acts faithfully ong{\displaystyle {\mathfrak {g}}}. The subgroup generated by the identity componentKof the unitary structure group and the operators in SU(2)m. It defines a compact Lie groupHwhich acts transitively onX. ThusH/Kis the corresponding Hermitian symmetric space of compact type. The groupGcan be identified with thecomplexificationofH. The subgroupH* leavingDinvariant is a noncompact real form ofG. It acts transitively onDso thatH* /Kis the dual Hermitian symmetric space of noncompact type. The inclusionsD⊂A⊂Xreproduce the Borel and Harish-Chandra embeddings. The classification of Hermitian symmetric spaces of tube type reduces to that of simple Euclidean Jordan algebras. These were classified byJordan, von Neumann & Wigner (1934)in terms ofEuclidean Hurwitz algebras, a special type ofcomposition algebra.
In general a Hermitian symmetric space gives rise to a 3-graded Lie algebra with a period 2 conjugate linear automorphism switching the parts of degree ±1 and preserving the degree 0 part. This gives rise to the structure of aJordan pairor hermitianJordan triple system, to whichLoos (1977)extended the theory of Jordan algebras. All irreducible Hermitian symmetric spaces can be constructed uniformly within this framework.Koecher (1969)constructed the irreducible Hermitian symmetric space of non-tube type from a simple Euclidean Jordan algebra together with a period 2 automorphism. The −1 eigenspace of the automorphism has the structure of a Jordan pair, which can be deduced from that of the larger Jordan algebra. In the non-tube type case corresponding to aSiegel domainof type II, there is no distinguished subgroup of real or complex Möbius transformations. For irreducible Hermitian symmetric spaces, tube type is characterized by the real dimension of the Shilov boundarySbeing equal to the complex dimension ofD.
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https://en.wikipedia.org/wiki/Hermitian_symmetric_space#Classical_domains
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Inmathematics, areal intervalis thesetof allreal numberslying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negativeinfinity, indicating the interval extends without abound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite.
For example, the set of real numbers consisting of0,1, and all numbers in between is an interval, denoted[0, 1]and called theunit interval; the set of allpositive real numbersis an interval, denoted(0, ∞); the set of all real numbers is an interval, denoted(−∞, ∞); and any single real numberais an interval, denoted[a,a].
Intervals are ubiquitous inmathematical analysis. For example, they occur implicitly in theepsilon-delta definition of continuity; theintermediate value theoremasserts that the image of an interval by acontinuous functionis an interval;integralsofreal functionsare defined over an interval; etc.
Interval arithmeticconsists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties ofinput dataandrounding errors.
Intervals are likewise defined on an arbitrarytotally orderedset, such asintegersorrational numbers. The notation of integer intervals is consideredin the special section below.
Anintervalis asubsetof thereal numbersthat contains all real numbers lying between any two numbers of the subset. In particular, theempty set∅{\displaystyle \varnothing }and the entire set of real numbersR{\displaystyle \mathbb {R} }are both intervals.
Theendpointsof an interval are itssupremum, and itsinfimum, if they exist as real numbers.[1]If the infimum does not exist, one says often that the corresponding endpoint is−∞.{\displaystyle -\infty .}Similarly, if the supremum does not exist, one says that the corresponding endpoint is+∞.{\displaystyle +\infty .}
Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of theleast-upper-bound propertyof the real numbers. This characterization is used to specify intervals by mean ofinterval notation, which is described below.
Anopen intervaldoes not include any endpoint, and is indicated with parentheses.[2]For example,(0,1)={x∣0<x<1}{\displaystyle (0,1)=\{x\mid 0<x<1\}}is the interval of all real numbers greater than0and less than1. (This interval can also be denoted by]0, 1[, see below). The open interval(0, +∞)consists of real numbers greater than0, i.e., positive real numbers. The open intervals have thus one of the forms
wherea{\displaystyle a}andb{\displaystyle b}are real numbers such thata<b.{\displaystyle a<b.}In the last case, the resulting interval is theempty setand does not depend ona{\displaystyle a}. The open intervals are those intervals that areopen setsfor the usualtopologyon the real numbers.
Aclosed intervalis an interval that includes all its endpoints and is denoted with square brackets.[2]For example,[0, 1]means greater than or equal to0and less than or equal to1. Closed intervals have one of the following forms in whichaandbare real numbers such thata<b:{\displaystyle a<b\colon }
The closed intervals are those intervals that areclosed setsfor the usualtopologyon the real numbers.
Ahalf-open intervalhas two endpoints and includes only one of them. It is saidleft-openorright-opendepending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals.[3]For example,(0, 1]means greater than0and less than or equal to1, while[0, 1)means greater than or equal to0and less than1. The half-open intervals have the form
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. The only intervals that appear twice in the above classification are∅{\displaystyle \emptyset }andR{\displaystyle \mathbb {R} }that are both open and closed.[4][5]
Adegenerate intervalis anyset consisting of a single real number(i.e., an interval of the form[a,a]).[6]Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to beproper, and has infinitely many elements.
An interval is said to beleft-boundedorright-bounded, if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to bebounded, if it is both left- and right-bounded; and is said to beunboundedotherwise. Intervals that are bounded at only one end are said to behalf-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known asfinite intervals.
Bounded intervals arebounded sets, in the sense that theirdiameter(which is equal to theabsolute differencebetween the endpoints) is finite. The diameter may be called thelength,width,measure,range, orsizeof the interval. The size of unbounded intervals is usually defined as+∞, and the size of the empty interval may be defined as0(or left undefined).
Thecentre(midpoint) of a bounded interval with endpointsaandbis(a+b)/2, and itsradiusis the half-length|a−b|/2. These concepts are undefined for empty or unbounded intervals.
An interval is said to beleft-openif and only if it contains nominimum(an element that is smaller than all other elements);right-openif it contains nomaximum; andopenif it contains neither. The interval[0, 1)= {x| 0 ≤x< 1}, for example, is left-closed and right-open. The empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals areopen setsof the real line in its standardtopology, and form abaseof the open sets.
An interval is said to beleft-closedif it has a minimum element or is left-unbounded,right-closedif it has a maximum or is right unbounded; it is simplyclosedif it is both left-closed and right closed. So, the closed intervals coincide with theclosed setsin that topology.
Theinteriorof an intervalIis the largest open interval that is contained inI; it is also the set of points inIwhich are not endpoints ofI. TheclosureofIis the smallest closed interval that containsI; which is also the setIaugmented with its finite endpoints.
For any setXof real numbers, theinterval enclosureorinterval spanofXis the unique interval that containsX, and does not properly contain any other interval that also containsX.
An intervalIis asubintervalof intervalJifIis asubsetofJ. An intervalIis aproper subintervalofJifIis aproper subsetofJ.
However, there is conflicting terminology for the termssegmentandinterval, which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. TheEncyclopedia of Mathematics[7]definesinterval(without a qualifier) to exclude both endpoints (i.e., open interval) andsegmentto include both endpoints (i.e., closed interval), while Rudin'sPrinciples of Mathematical Analysis[8]calls sets of the form [a,b]intervalsand sets of the form (a,b)segmentsthroughout. These terms tend to appear in older works; modern texts increasingly favor the terminterval(qualified byopen,closed, orhalf-open), regardless of whether endpoints are included.
The interval of numbers betweenaandb, includingaandb, is often denoted[a,b]. The two numbers are called theendpointsof the interval. In countries where numbers are written with adecimal comma, asemicolonmay be used as a separator to avoid ambiguity.
To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described inInternational standardISO 31-11. Thus, inset builder notation,
Each interval(a,a),[a,a), and(a,a]represents theempty set, whereas[a,a]denotes the singleton set{a}. Whena>b, all four notations are usually taken to represent the empty set.
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation(a,b)is often used to denote anordered pairin set theory, thecoordinatesof apointorvectorinanalytic geometryandlinear algebra, or (sometimes) acomplex numberinalgebra. That is whyBourbakiintroduced the notation]a,b[to denote the open interval.[9]The notation[a,b]too is occasionally used for ordered pairs, especially incomputer science.
Some authors such as Yves Tillé use]a,b[to denote the complement of the interval(a,b); namely, the set of all real numbers that are either less than or equal toa, or greater than or equal tob.
In some contexts, an interval may be defined as a subset of theextended real numbers, the set of all real numbers augmented with−∞and+∞.
In this interpretation, the notations[−∞,b],(−∞,b],[a, +∞], and[a, +∞)are all meaningful and distinct. In particular,(−∞, +∞)denotes the set of all ordinary real numbers, while[−∞, +∞]denotes the extended reals.
Even in the context of the ordinary reals, one may use aninfiniteendpoint to indicate that there is no bound in that direction. For example,(0, +∞)is the set ofpositive real numbers, also written asR+.{\displaystyle \mathbb {R} _{+}.}The context affects some of the above definitions and terminology. For instance, the interval(−∞, +∞)=R{\displaystyle \mathbb {R} }is closed in the realm of ordinary reals, but not in the realm of the extended reals.
Whenaandbareintegers, the notation ⟦a, b⟧, or[a..b]or{a..b}or justa..b, is sometimes used to indicate the interval of allintegersbetweenaandbincluded. The notation[a..b]is used in someprogramming languages; inPascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of validindicesof anarray.
Another way to interpret integer intervals are assets defined by enumeration, usingellipsisnotation.
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writinga..b− 1,a+ 1 ..b, ora+ 1 ..b− 1. Alternate-bracket notations like[a..b)or[a..b[are rarely used for integer intervals.[citation needed]
The intervals are precisely theconnectedsubsets ofR.{\displaystyle \mathbb {R} .}It follows that the image of an interval by anycontinuous functionfromR{\displaystyle \mathbb {R} }toR{\displaystyle \mathbb {R} }is also an interval. This is one formulation of theintermediate value theorem.
The intervals are also theconvex subsetsofR.{\displaystyle \mathbb {R} .}The interval enclosure of a subsetX⊆R{\displaystyle X\subseteq \mathbb {R} }is also theconvex hullofX.{\displaystyle X.}
Theclosureof an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of everyconnected subsetof atopological spaceis a connected subset.) In other words, we have[10]
The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example(a,b)∪[b,c]=(a,c].{\displaystyle (a,b)\cup [b,c]=(a,c].}
IfR{\displaystyle \mathbb {R} }is viewed as ametric space, itsopen ballsare the open bounded intervals(c+r,c−r), and itsclosed ballsare the closed bounded intervals[c+r,c−r]. In particular, themetricandordertopologies in the real line coincide, which is the standard topology of the real line.
Any elementxof an intervalIdefines a partition ofIinto three disjoint intervalsI1,I2,I3: respectively, the elements ofIthat are less thanx, the singleton[x,x]={x},{\displaystyle [x,x]=\{x\},}and the elements that are greater thanx. The partsI1andI3are both non-empty (and have non-empty interiors), if and only ifxis in the interior ofI. This is an interval version of thetrichotomy principle.
Adyadic intervalis a bounded real interval whose endpoints arej2n{\displaystyle {\tfrac {j}{2^{n}}}}andj+12n,{\displaystyle {\tfrac {j+1}{2^{n}}},}wherej{\displaystyle j}andn{\displaystyle n}are integers. Depending on the context, either endpoint may or may not be included in the interval.
Dyadic intervals have the following properties:
The dyadic intervals consequently have a structure that reflects that of an infinitebinary tree.
Dyadic intervals are relevant to several areas of numerical analysis, includingadaptive mesh refinement,multigrid methodsandwavelet analysis. Another way to represent such a structure isp-adic analysis(forp= 2).[11]
An open finite interval(a,b){\displaystyle (a,b)}is a 1-dimensional openballwith acenterat12(a+b){\displaystyle {\tfrac {1}{2}}(a+b)}and aradiusof12(b−a).{\displaystyle {\tfrac {1}{2}}(b-a).}The closed finite interval[a,b]{\displaystyle [a,b]}is the corresponding closed ball, and the interval's two endpoints{a,b}{\displaystyle \{a,b\}}form a 0-dimensionalsphere. Generalized ton{\displaystyle n}-dimensionalEuclidean space, a ball is the set of points whose distance from the center is less than the radius. In the 2-dimensional case, a ball is called adisk.
If ahalf-spaceis taken as a kind ofdegenerateball (without a well-defined center or radius), a half-space can be taken as analogous to a half-bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint.
A finite interval is (the interior of) a 1-dimensionalhyperrectangle. Generalized toreal coordinate spaceRn,{\displaystyle \mathbb {R} ^{n},}anaxis-alignedhyperrectangle (or box) is theCartesian productofn{\displaystyle n}finite intervals. Forn=2{\displaystyle n=2}this is arectangle; forn=3{\displaystyle n=3}this is arectangular cuboid(also called a "box").
Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of anyn{\displaystyle n}intervals,I=I1×I2×⋯×In{\displaystyle I=I_{1}\times I_{2}\times \cdots \times I_{n}}is sometimes called ann{\displaystyle n}-dimensional interval.[citation needed]
Afacetof such an intervalI{\displaystyle I}is the result of replacing any non-degenerate interval factorIk{\displaystyle I_{k}}by a degenerate interval consisting of a finite endpoint ofIk.{\displaystyle I_{k}.}ThefacesofI{\displaystyle I}compriseI{\displaystyle I}itself and all faces of its facets. ThecornersofI{\displaystyle I}are the faces that consist of a single point ofRn.{\displaystyle \mathbb {R} ^{n}.}[citation needed]
Any finite interval can be constructed as theintersectionof half-bounded intervals (with an empty intersection taken to mean the whole real line), and the intersection of any number of half-bounded intervals is a (possibly empty) interval. Generalized ton{\displaystyle n}-dimensionalaffine space, an intersection of half-spaces (of arbitrary orientation) is (the interior of) aconvex polytope, or in the 2-dimensional case aconvex polygon.
An open interval is a connected open set of real numbers. Generalized totopological spacesin general, a non-empty connected open set is called adomain.
Intervals ofcomplex numberscan be defined as regions of thecomplex plane, eitherrectangularorcircular.[12]
The concept of intervals can be defined in arbitrarypartially ordered setsor more generally, in arbitrarypreordered sets. For apreordered set(X,≲){\displaystyle (X,\lesssim )}and two elementsa,b∈X,{\displaystyle a,b\in X,}one similarly defines the intervals[13]: 11, Definition 11
wherex<y{\displaystyle x<y}meansx≲y≴x.{\displaystyle x\lesssim y\not \lesssim x.}Actually, the intervals with single or no endpoints are the same as the intervals with two endpoints in the larger preordered set
defined by adding new smallest and greatest elements (even if there were ones), which are subsets ofX.{\displaystyle X.}In the case ofX=R{\displaystyle X=\mathbb {R} }one may takeR¯{\displaystyle {\bar {\mathbb {R} }}}to be theextended real line.
A subsetA⊆X{\displaystyle A\subseteq X}of thepreordered set(X,≲){\displaystyle (X,\lesssim )}is(order-)convexif for everyx,y∈A{\displaystyle x,y\in A}and everyx≲z≲y{\displaystyle x\lesssim z\lesssim y}we havez∈A.{\displaystyle z\in A.}Unlike in the case of the real line, a convex set of a preordered set need not be an interval. For example, in thetotally ordered set(Q,≤){\displaystyle (\mathbb {Q} ,\leq )}ofrational numbers, the set
is convex, but not an interval ofQ,{\displaystyle \mathbb {Q} ,}since there is no square root of two inQ.{\displaystyle \mathbb {Q} .}
Let(X,≲){\displaystyle (X,\lesssim )}be apreordered setand letY⊆X.{\displaystyle Y\subseteq X.}The convex sets ofX{\displaystyle X}contained inY{\displaystyle Y}form aposetunder inclusion. Amaximal elementof this poset is called aconvex componentofY.{\displaystyle Y.}[14]: Definition 5.1[15]: 727By theZorn lemma, any convex set ofX{\displaystyle X}contained inY{\displaystyle Y}is contained in some convex component ofY,{\displaystyle Y,}but such components need not be unique. In atotally ordered set, such a component is always unique. That is, the convex components of a subset of a totally ordered set form apartition.
A generalization of the characterizations of the real intervals follows. For a non-empty subsetI{\displaystyle I}of alinear continuum(L,≤),{\displaystyle (L,\leq ),}the following conditions are equivalent.[16]: 153, Theorem 24.1
For asubsetS{\displaystyle S}of alatticeL,{\displaystyle L,}the following conditions are equivalent.
EveryTychonoff spaceis embeddable into aproduct spaceof the closed unit intervals[0,1].{\displaystyle [0,1].}Actually, every Tychonoff space that has abaseofcardinalityκ{\displaystyle \kappa }is embeddable into the product[0,1]κ{\displaystyle [0,1]^{\kappa }}ofκ{\displaystyle \kappa }copies of the intervals.[17]: p. 83, Theorem 2.3.23
The concepts of convex sets and convex components are used in a proof that everytotally ordered setendowed with theorder topologyiscompletely normal[15]or moreover,monotonically normal.[14]
Intervals can be associated with points of the plane, and hence regions of intervals can be associated withregionsof the plane. Generally, an interval in mathematics corresponds to an ordered pair(x,y)taken from thedirect productR×R{\displaystyle \mathbb {R} \times \mathbb {R} }of real numbers with itself, where it is often assumed thaty>x. For purposes ofmathematical structure, this restriction is discarded,[18]and "reversed intervals" wherey−x< 0are allowed. Then, the collection of all intervals[x,y]can be identified with thetopological ringformed by thedirect sumofR{\displaystyle \mathbb {R} }with itself, where addition and multiplication are defined component-wise.
The direct sum algebra(R⊕R,+,×){\displaystyle (\mathbb {R} \oplus \mathbb {R} ,+,\times )}has twoideals, { [x,0] :x∈ R } and { [0,y] :y∈ R }. Theidentity elementof this algebra is the condensed interval[1, 1]. If interval[x,y]is not in one of the ideals, then it hasmultiplicative inverse[1/x, 1/y]. Endowed with the usualtopology, the algebra of intervals forms atopological ring. Thegroup of unitsof this ring consists of fourquadrantsdetermined by the axes, or ideals in this case. Theidentity componentof this group is quadrant I.
Every interval can be considered a symmetric interval around itsmidpoint. In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals"[x, −x]is used along with the axis of intervals[x,x]that reduce to a point. Instead of the direct sumR⊕R,{\displaystyle R\oplus R,}the ring of intervals has been identified[19]with thehyperbolic numbersby M. Warmus andD. H. Lehmerthrough the identification
wherej2=1.{\displaystyle j^{2}=1.}
This linear mapping of the plane, which amounts of aring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such aspolar decomposition.
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Inmathematics, aLipschitz domain(ordomain with Lipschitz boundary) is adomaininEuclidean spacewhoseboundaryis "sufficiently regular" in the sense that it can be thought of as locally being the graph of aLipschitz continuous function. The term is named after theGermanmathematicianRudolf Lipschitz.
Letn∈N{\displaystyle n\in \mathbb {N} }. LetΩ{\displaystyle \Omega }be adomainofRn{\displaystyle \mathbb {R} ^{n}}and let∂Ω{\displaystyle \partial \Omega }denote theboundaryofΩ{\displaystyle \Omega }. ThenΩ{\displaystyle \Omega }is called aLipschitz domainif for every pointp∈∂Ω{\displaystyle p\in \partial \Omega }there exists ahyperplaneH{\displaystyle H}of dimensionn−1{\displaystyle n-1}throughp{\displaystyle p}, a Lipschitz-continuous functiong:H→R{\displaystyle g:H\rightarrow \mathbb {R} }over that hyperplane, and realsr>0{\displaystyle r>0}andh>0{\displaystyle h>0}such that
where
In other words, at each point of its boundary,Ω{\displaystyle \Omega }is locally the set of points located above the graph of some Lipschitz function.
A more general notion is that ofweakly Lipschitzdomains, which are domains whose boundary is locally flattable by a bilipschitz mapping. Lipschitz domains in the sense above are sometimes calledstrongly Lipschitzby contrast with weakly Lipschitz domains.
A domainΩ{\displaystyle \Omega }isweakly Lipschitzif for every pointp∈∂Ω,{\displaystyle p\in \partial \Omega ,}there exists a radiusr>0{\displaystyle r>0}and a mapℓp:Br(p)→Q{\displaystyle \ell _{p}:B_{r}(p)\rightarrow Q}such that
whereQ{\displaystyle Q}denotes the unit ballB1(0){\displaystyle B_{1}(0)}inRn{\displaystyle \mathbb {R} ^{n}}and
A (strongly) Lipschitz domain is always a weakly Lipschitz domain but the converse is not true. An example of weakly Lipschitz domains that fails to be a strongly Lipschitz domain is given by thetwo-bricksdomain[1]
Many of theSobolev embedding theoremsrequire that the domain of study be a Lipschitz domain. Consequently, manypartial differential equationsandvariational problemsare defined on Lipschitz domains.
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Inmathematics,point-free geometryis ageometrywhose primitiveontologicalnotion isregionrather thanpoint. Twoaxiomatic systemsare set out below, one grounded inmereology, the other inmereotopologyand known asconnection theory.
Point-free geometry was first formulated byAlfred North Whitehead,[1]not as a theory ofgeometryor ofspacetime, but of "events" and of an "extensionrelation" between events. Whitehead's purposes were as muchphilosophicalas scientific and mathematical.[2]
Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formalfirst-order theoriesdescribed in this entry were devised by others in order to clarify and refine Whitehead's theories. Thedomain of discoursefor both theories consists of "regions." Allunquantifiedvariables in this entry should be taken as tacitlyuniversally quantified; hence all axioms should be taken asuniversal closures. No axiom requires more than three quantified variables; hence a translation of first-order theories intorelation algebrais possible. Each set of axioms has but fourexistential quantifiers.
The fundamental primitivebinary relationisinclusion, denoted by theinfix operator"≤", which corresponds to the binaryParthoodrelation that is a standard feature inmereologicaltheories. The intuitive meaning ofx≤yis "xis part ofy." Assuming that equality, denoted by the infix operator "=", is part of the background logic, the binary relationProper Part, denoted by the infix operator "<", is defined as:
The axioms are:[3]
AmodelofG1–G7is aninclusion space.
Definition.[4]Given some inclusion space S, anabstractive classis a classGof regions such thatS\Gistotally orderedby inclusion. Moreover, there does not exist a region included in all of the regions included inG.
Intuitively, an abstractive class defines a geometrical entity whose dimensionality is less than that of the inclusion space. For example, if the inclusion space is theEuclidean plane, then the corresponding abstractive classes arepointsandlines.
Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's systemW.[5]In turn,Wformalizes a theory of Whitehead[6]whose axioms are not made explicit. Point-free geometry isWwith this defect repaired. Simons did not repair this defect, instead proposing in a footnote that the reader do so as an exercise. The primitive relation ofWis Proper Part, astrict partial order. The theory[7]of Whitehead (1919) has a single primitive binary relationKdefined asxKy↔y<x. HenceKis theconverseof Proper Part. Simons'sWP1asserts that Proper Part isirreflexiveand so corresponds toG1.G3establishes that inclusion, unlike Proper Part, isantisymmetric.
Point-free geometry is closely related to adense linear orderD, whose axioms areG1-3,G5, and the totality axiomx≤y∨y≤x.{\displaystyle x\leq y\lor y\leq x.}[8]Hence inclusion-based point-free geometry would be a proper extension ofD(namelyD∪ {G4,G6,G7}), were it not that theDrelation "≤" is atotal order.
A different approach was proposed in Whitehead (1929), one inspired by De Laguna (1922). Whitehead took as primitive thetopologicalnotion of "contact" between two regions, resulting in a primitive "connection relation" between events. Connection theoryCis afirst-order theorythat distills the first 12 of Whitehead's 31 assumptions[9]into 6 axioms,C1-C6.[10]Cis a proper fragment of the theories proposed by Clarke,[11]who noted theirmereologicalcharacter. Theories that, likeC, feature both inclusion and topological primitives, are calledmereotopologies.
Chas one primitiverelation, binary "connection," denoted by theprefixedpredicate letterC. Thatxis included inycan now be defined asx≤y↔ ∀z[Czx→Czy]. Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion,[12]a total order that enables the construction of abstractive classes. Gerla and Miranda (2008) argue that only thus can mereotopology unambiguously define apoint.
A model ofCis aconnection space.
Following the verbal description of each axiom is the identifier of the corresponding axiom in Casati and Varzi (1999). Their systemSMT(strong mereotopology) consists ofC1-C3, and is essentially due to Clarke (1981).[13]Any mereotopology can be madeatomlessby invokingC4, without risking paradox or triviality. HenceCextends the atomless variant ofSMTby means of the axiomsC5andC6, suggested by chapter 2 of part 4 ofProcess and Reality.[14]
Biacino and Gerla (1991) showed that everymodelof Clarke's theory is aBoolean algebra, and models of such algebras cannot distinguish connection from overlap. It is doubtful whether either fact is faithful to Whitehead's intent.
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Inmathematics, abase(orbasis;pl.:bases) for thetopologyτof atopological space(X, τ)is afamilyB{\displaystyle {\mathcal {B}}}ofopen subsetsofXsuch that every open set of the topology is equal to theunionof somesub-familyofB{\displaystyle {\mathcal {B}}}. For example, the set of allopen intervalsin thereal number lineR{\displaystyle \mathbb {R} }is a basis for theEuclidean topologyonR{\displaystyle \mathbb {R} }because every open interval is an open set, and also every open subset ofR{\displaystyle \mathbb {R} }can be written as a union of some family of open intervals.
Bases are ubiquitous throughout topology. The sets in a base for a topology, which are calledbasic open sets, are often easier to describe and use than arbitrary open sets.[1]Many important topological definitions such ascontinuityandconvergencecan be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.
Not all families of subsets of a setX{\displaystyle X}form a base for a topology onX{\displaystyle X}. Under some conditions detailed below, a family of subsets will form a base for a (unique) topology onX{\displaystyle X}, obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of asubbasefor a topology. Bases for topologies are also closely related toneighborhood bases.
Given atopological space(X,τ){\displaystyle (X,\tau )}, abase[2](orbasis[3]) for thetopologyτ{\displaystyle \tau }(also called abase forX{\displaystyle X}if the topology is understood) is afamilyB⊆τ{\displaystyle {\mathcal {B}}\subseteq \tau }of open sets such that every open set of the topology can be represented as the union of some subfamily ofB{\displaystyle {\mathcal {B}}}.[note 1]The elements ofB{\displaystyle {\mathcal {B}}}are calledbasic open sets.
Equivalently, a familyB{\displaystyle {\mathcal {B}}}of subsets ofX{\displaystyle X}is a base for the topologyτ{\displaystyle \tau }if and only ifB⊆τ{\displaystyle {\mathcal {B}}\subseteq \tau }and for every open setU{\displaystyle U}inX{\displaystyle X}and pointx∈U{\displaystyle x\in U}there is some basic open setB∈B{\displaystyle B\in {\mathcal {B}}}such thatx∈B⊆U{\displaystyle x\in B\subseteq U}.
For example, the collection of allopen intervalsin thereal lineforms a base for the standard topology on the real numbers. More generally, in a metric spaceM{\displaystyle M}the collection of all open balls about points ofM{\displaystyle M}forms a base for the topology.
In general, a topological space(X,τ){\displaystyle (X,\tau )}can have many bases. The whole topologyτ{\displaystyle \tau }is always a base for itself (that is,τ{\displaystyle \tau }is a base forτ{\displaystyle \tau }). For the real line, the collection of all open intervals is a base for the topology. So is the collection of all open intervals with rational endpoints, or the collection of all open intervals with irrational endpoints, for example. Note that two different bases need not have any basic open set in common. One of thetopological propertiesof a spaceX{\displaystyle X}is the minimumcardinalityof a base for its topology, called theweightofX{\displaystyle X}and denotedw(X){\displaystyle w(X)}. From the examples above, the real line has countable weight.
IfB{\displaystyle {\mathcal {B}}}is a base for the topologyτ{\displaystyle \tau }of a spaceX{\displaystyle X}, it satisfies the following properties:[4]
Property (B1) corresponds to the fact thatX{\displaystyle X}is an open set; property (B2) corresponds to the fact thatB1∩B2{\displaystyle B_{1}\cap B_{2}}is an open set.
Conversely, supposeX{\displaystyle X}is just a set without any topology andB{\displaystyle {\mathcal {B}}}is a family of subsets ofX{\displaystyle X}satisfying properties (B1) and (B2). ThenB{\displaystyle {\mathcal {B}}}is a base for the topology that it generates. More precisely, letτ{\displaystyle \tau }be the family of all subsets ofX{\displaystyle X}that are unions of subfamilies ofB.{\displaystyle {\mathcal {B}}.}Thenτ{\displaystyle \tau }is a topology onX{\displaystyle X}andB{\displaystyle {\mathcal {B}}}is a base forτ{\displaystyle \tau }.[5](Sketch:τ{\displaystyle \tau }defines a topology because it is stable under arbitrary unions by construction, it is stable under finite intersections by (B2), it containsX{\displaystyle X}by (B1), and it contains the empty set as the union of the empty subfamily ofB{\displaystyle {\mathcal {B}}}. The familyB{\displaystyle {\mathcal {B}}}is then a base forτ{\displaystyle \tau }by construction.) Such families of sets are a very common way of defining a topology.
In general, ifX{\displaystyle X}is a set andB{\displaystyle {\mathcal {B}}}is an arbitrary collection of subsets ofX{\displaystyle X}, there is a (unique) smallest topologyτ{\displaystyle \tau }onX{\displaystyle X}containingB{\displaystyle {\mathcal {B}}}. (This topology is theintersectionof all topologies onX{\displaystyle X}containingB{\displaystyle {\mathcal {B}}}.) The topologyτ{\displaystyle \tau }is called thetopology generated byB{\displaystyle {\mathcal {B}}}, andB{\displaystyle {\mathcal {B}}}is called asubbaseforτ{\displaystyle \tau }.
The topologyτ{\displaystyle \tau }consists ofX{\displaystyle X}together with all arbitrary unions of finite intersections of elements ofB{\displaystyle {\mathcal {B}}}(see the article aboutsubbase.) Now, ifB{\displaystyle {\mathcal {B}}}also satisfies properties (B1) and (B2), the topology generated byB{\displaystyle {\mathcal {B}}}can be described in a simpler way without having to take intersections:τ{\displaystyle \tau }is the set of all unions of elements ofB{\displaystyle {\mathcal {B}}}(andB{\displaystyle {\mathcal {B}}}is a base forτ{\displaystyle \tau }in that case).
There is often an easy way to check condition (B2). If the intersection of any two elements ofB{\displaystyle {\mathcal {B}}}is itself an element ofB{\displaystyle {\mathcal {B}}}or is empty, then condition (B2) is automatically satisfied (by takingB3=B1∩B2{\displaystyle B_{3}=B_{1}\cap B_{2}}). For example, theEuclidean topologyon the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.
An example of a collection of open sets that is not a base is the setS{\displaystyle S}of all semi-infinite intervals of the forms(−∞,a){\displaystyle (-\infty ,a)}and(a,∞){\displaystyle (a,\infty )}witha∈R{\displaystyle a\in \mathbb {R} }. The topology generated byS{\displaystyle S}contains all open intervals(a,b)=(−∞,b)∩(a,∞){\displaystyle (a,b)=(-\infty ,b)\cap (a,\infty )}, henceS{\displaystyle S}generates the standard topology on the real line. ButS{\displaystyle S}is only a subbase for the topology, not a base: a finite open interval(a,b){\displaystyle (a,b)}does not contain any element ofS{\displaystyle S}(equivalently, property (B2) does not hold).
The setΓof all open intervals inR{\displaystyle \mathbb {R} }forms a basis for theEuclidean topologyonR{\displaystyle \mathbb {R} }.
A non-empty family of subsets of a setXthat is closed under finite intersections of two or more sets, which is called aπ-systemonX, is necessarily a base for a topology onXif and only if it coversX. By definition, everyσ-algebra, everyfilter(and so in particular, everyneighborhood filter), and everytopologyis a coveringπ-system and so also a base for a topology. In fact, ifΓis a filter onXthen{ ∅ } ∪ Γis a topology onXandΓis a basis for it. A base for a topology does not have to be closed under finite intersections and many are not. But nevertheless, many topologies are defined by bases that are also closed under finite intersections. For example, each of the following families of subset ofR{\displaystyle \mathbb {R} }is closed under finite intersections and so each forms a basis forsometopology onR{\displaystyle \mathbb {R} }:
TheZariski topologyon thespectrum of a ringhas a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set.
Closed setsare equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological spaceX,{\displaystyle X,}afamilyC{\displaystyle {\mathcal {C}}}of closed sets forms abase for the closed setsif and only if for each closed setA{\displaystyle A}and each pointx{\displaystyle x}not inA{\displaystyle A}there exists an element ofC{\displaystyle {\mathcal {C}}}containingA{\displaystyle A}but not containingx.{\displaystyle x.}A familyC{\displaystyle {\mathcal {C}}}is a base for the closed sets ofX{\displaystyle X}if and only if itsdualinX,{\displaystyle X,}that is the family{X∖C:C∈C}{\displaystyle \{X\setminus C:C\in {\mathcal {C}}\}}ofcomplementsof members ofC{\displaystyle {\mathcal {C}}}, is a base for the open sets ofX.{\displaystyle X.}
LetC{\displaystyle {\mathcal {C}}}be a base for the closed sets ofX.{\displaystyle X.}Then
Any collection of subsets of a setX{\displaystyle X}satisfying these properties forms a base for the closed sets of a topology onX.{\displaystyle X.}The closed sets of this topology are precisely the intersections of members ofC.{\displaystyle {\mathcal {C}}.}
In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space iscompletely regularif and only if thezero setsform a base for the closed sets. Given any topological spaceX,{\displaystyle X,}the zero sets form the base for the closed sets of some topology onX.{\displaystyle X.}This topology will be the finest completely regular topology onX{\displaystyle X}coarser than the original one. In a similar vein, theZariski topologyonAnis defined by taking the zero sets of polynomial functions as a base for the closed sets.
We shall work with notions established in (Engelking 1989, p. 12, pp. 127-128).
FixX{\displaystyle X}a topological space. Here, anetworkis a familyN{\displaystyle {\mathcal {N}}}of sets, for which, for all pointsx{\displaystyle x}and open neighbourhoodsUcontainingx{\displaystyle x}, there existsB{\displaystyle B}inN{\displaystyle {\mathcal {N}}}for whichx∈B⊆U.{\displaystyle x\in B\subseteq U.}Note that, unlike a basis, the sets in a network need not be open.
We define theweight,w(X){\displaystyle w(X)}, as the minimum cardinality of a basis; we define thenetwork weight,nw(X){\displaystyle nw(X)}, as the minimum cardinality of a network; thecharacter of a point,χ(x,X),{\displaystyle \chi (x,X),}as the minimum cardinality of a neighbourhood basis forx{\displaystyle x}inX{\displaystyle X}; and thecharacterofX{\displaystyle X}to beχ(X)≜sup{χ(x,X):x∈X}.{\displaystyle \chi (X)\triangleq \sup\{\chi (x,X):x\in X\}.}
The point of computing the character and weight is to be able to tell what sort of bases and local bases can exist. We have the following facts:
The last fact follows fromf(X){\displaystyle f(X)}being compact Hausdorff, and hencenw(f(X))=w(f(X))≤w(X)≤ℵ0{\displaystyle nw(f(X))=w(f(X))\leq w(X)\leq \aleph _{0}}(since compact metrizable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrizable exactly in case they are second countable. (An application of this, for instance, is that every path in a Hausdorff space is compact metrizable.)
Using the above notation, suppose thatw(X)≤κ{\displaystyle w(X)\leq \kappa }some infinite cardinal. Then there does not exist a strictly increasing sequence of open sets (equivalently strictly decreasing sequence of closed sets) of length≤κ+{\displaystyle \leq \kappa ^{+}\!}.
To see this (without the axiom of choice), fix{Uξ}ξ∈κ,{\displaystyle \left\{U_{\xi }\right\}_{\xi \in \kappa },}as a basis of open sets. And supposeper contra, that{Vξ}ξ∈κ+{\displaystyle \left\{V_{\xi }\right\}_{\xi \in \kappa ^{+}}}were a strictly increasing sequence of open sets. This means∀α<κ+:Vα∖⋃ξ<αVξ≠∅.{\displaystyle \forall \alpha <\kappa ^{+}\!:\qquad V_{\alpha }\setminus \bigcup _{\xi <\alpha }V_{\xi }\neq \varnothing .}
Forx∈Vα∖⋃ξ<αVξ,{\displaystyle x\in V_{\alpha }\setminus \bigcup _{\xi <\alpha }V_{\xi },}we may use the basis to find someUγ{\displaystyle U_{\gamma }}withx{\displaystyle x}inUγ⊆Vα{\displaystyle U_{\gamma }\subseteq V_{\alpha }}. In this way we may well-define a map,f:κ+→κ{\displaystyle f:\kappa ^{+}\!\to \kappa }mapping eachα{\displaystyle \alpha }to the leastγ{\displaystyle \gamma }for whichUγ⊆Vα{\displaystyle U_{\gamma }\subseteq V_{\alpha }}and meetsVα∖⋃ξ<αVξ.{\displaystyle V_{\alpha }\setminus \bigcup _{\xi <\alpha }V_{\xi }.}
This map is injective, otherwise there would beα<β{\displaystyle \alpha <\beta }withf(α)=f(β)=γ{\displaystyle f(\alpha )=f(\beta )=\gamma }, which would further implyUγ⊆Vα{\displaystyle U_{\gamma }\subseteq V_{\alpha }}but also meetsVβ∖⋃ξ<αVξ⊆Vβ∖Vα,{\displaystyle V_{\beta }\setminus \bigcup _{\xi <\alpha }V_{\xi }\subseteq V_{\beta }\setminus V_{\alpha },}which is a contradiction. But this would go to show thatκ+≤κ{\displaystyle \kappa ^{+}\!\leq \kappa }, a contradiction.
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Inmathematical analysis, adomainorregionis anon-empty,connected, andopen setin atopological space. In particular, it is any non-empty connected opensubsetof thereal coordinate spaceRnor thecomplex coordinate spaceCn. A connected open subset ofcoordinate spaceis frequently used for thedomain of a function.[1]
The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the termdomain,[2]some use the termregion,[3]some use both terms interchangeably,[4]and some define the two terms slightly differently;[5]some avoid ambiguity by sticking with a phrase such asnon-empty connected open subset.[6]
One common convention is to define adomainas a connected open set but aregionas theunionof a domain with none, some, or all of itslimit points.[7]Aclosed regionorclosed domainis the union of a domain and all of its limit points.
Various degrees of smoothness of theboundaryof the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (Green's theorem,Stokes theorem), properties ofSobolev spaces, and to definemeasureson the boundary and spaces oftraces(generalized functions defined on the boundary). Commonly considered types of domains are domains withcontinuousboundary,Lipschitz boundary,C1boundary, and so forth.
Abounded domainis a domain that isbounded, i.e., contained in some ball.Bounded regionis defined similarly. Anexterior domainorexternal domainis a domain whosecomplementis bounded; sometimes smoothness conditions are imposed on its boundary.
Incomplex analysis, acomplex domain(or simplydomain) is any connected open subset of thecomplex planeC. For example, the entire complex plane is a domain, as is the openunit disk, the openupper half-plane, and so forth. Often, a complex domain serves as thedomain of definitionfor aholomorphic function. In the study ofseveral complex variables, the definition of a domain is extended to include any connected open subset ofCn.
InEuclidean spaces,one-,two-, andthree-dimensionalregions arecurves,surfaces, andsolids, whose extent are called, respectively,length,area, andvolume.
Definition. An open set is connected if it cannot be expressed as the sum of two open sets. An open connected set is called a domain.
German:Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.
According toHans Hahn,[8]the concept of a domain as an open connected set was introduced byConstantin Carathéodoryin his famous book (Carathéodory 1918).
In this definition, Carathéodory considers obviouslynon-emptydisjointsets.
Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as asynonymofopen set.[9]The rough concept is older. In the 19th and early 20th century, the termsdomainandregionwere often used informally (sometimes interchangeably) without explicit definition.[10]
However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influentialmonographsonelliptic partial differential equations,Carlo Mirandauses the term "region" to identify an open connected set,[11][12]and reserves the term "domain" to identify an internally connected,[13]perfect set, each point of which is an accumulation point of interior points,[11]following his former masterMauro Picone:[14]according to this convention, if a setAis a region then itsclosureAis a domain.[11]
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https://en.wikipedia.org/wiki/Domain_(mathematical_analysis)
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Inmathematics, more specifically intopology, anopen mapis afunctionbetween twotopological spacesthat mapsopen setsto open sets.[1][2][3]That is, a functionf:X→Y{\displaystyle f:X\to Y}is open if for any open setU{\displaystyle U}inX,{\displaystyle X,}theimagef(U){\displaystyle f(U)}is open inY.{\displaystyle Y.}Likewise, aclosed mapis a function that mapsclosed setsto closed sets.[3][4]A map may be open, closed, both, or neither;[5]in particular, an open map need not be closed and vice versa.[6]
Open[7]and closed[8]maps are not necessarilycontinuous.[4]Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property;[3]this fact remains true even if one restricts oneself to metric spaces.[9]Although their definitions seem more natural, open and closed maps are much less important than continuous maps.
Recall that, by definition, a functionf:X→Y{\displaystyle f:X\to Y}is continuous if thepreimageof every open set ofY{\displaystyle Y}is open inX.{\displaystyle X.}[2](Equivalently, if the preimage of every closed set ofY{\displaystyle Y}is closed inX{\displaystyle X}).
Early study of open maps was pioneered bySimion StoilowandGordon Thomas Whyburn.[10]
IfS{\displaystyle S}is a subset of a topological space then letS¯{\displaystyle {\overline {S}}}andClS{\displaystyle \operatorname {Cl} S}(resp.IntS{\displaystyle \operatorname {Int} S}) denote theclosure(resp.interior) ofS{\displaystyle S}in that space.
Letf:X→Y{\displaystyle f:X\to Y}be a function betweentopological spaces. IfS{\displaystyle S}is any set thenf(S):={f(s):s∈S∩domainf}{\displaystyle f(S):=\left\{f(s)~:~s\in S\cap \operatorname {domain} f\right\}}is called the image ofS{\displaystyle S}underf.{\displaystyle f.}
There are two different competing, but closely related, definitions of "open map" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets."
The following terminology is sometimes used to distinguish between the two definitions.
A mapf:X→Y{\displaystyle f:X\to Y}is called a
Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.
Asurjectivemap is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent.
More generally, a mapf:X→Y{\displaystyle f:X\to Y}is relatively open if and only if thesurjectionf:X→f(X){\displaystyle f:X\to f(X)}is a strongly open map.
BecauseX{\displaystyle X}is always an open subset ofX,{\displaystyle X,}the imagef(X)=Imf{\displaystyle f(X)=\operatorname {Im} f}of a strongly open mapf:X→Y{\displaystyle f:X\to Y}must be an open subset of its codomainY.{\displaystyle Y.}In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain.
In summary,
By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition.
The discussion above will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".
A mapf:X→Y{\displaystyle f:X\to Y}is called anopen mapor astrongly open mapif it satisfies any of the following equivalent conditions:
IfB{\displaystyle {\mathcal {B}}}is abasisforX{\displaystyle X}then the following can be appended to this list:
A mapf:X→Y{\displaystyle f:X\to Y}is called arelatively closed mapif wheneverC{\displaystyle C}is aclosed subsetof the domainX{\displaystyle X}thenf(C){\displaystyle f(C)}is a closed subset off{\displaystyle f}'simageImf:=f(X),{\displaystyle \operatorname {Im} f:=f(X),}where as usual, this set is endowed with thesubspace topologyinduced on it byf{\displaystyle f}'scodomainY.{\displaystyle Y.}
A mapf:X→Y{\displaystyle f:X\to Y}is called aclosed mapor astrongly closed mapif it satisfies any of the following equivalent conditions:
Asurjectivemap is strongly closed if and only if it is relatively closed. So for this important special case, the two definitions are equivalent.
By definition, the mapf:X→Y{\displaystyle f:X\to Y}is a relatively closed map if and only if thesurjectionf:X→Imf{\displaystyle f:X\to \operatorname {Im} f}is a strongly closed map.
If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") isequivalentto continuity.
This does not happen with the definition of "open map" (which is: "every image of an open set is open") since the statement that results ("every image of a closed set is closed") is the definition of "closed map", which is in generalnotequivalent to openness. There exist open maps that are not closed and there also exist closed maps that are not open. This difference between open/closed maps and continuous maps is ultimately due to the fact that for any setS,{\displaystyle S,}onlyf(X∖S)⊇f(X)∖f(S){\displaystyle f(X\setminus S)\supseteq f(X)\setminus f(S)}is guaranteed in general, whereas for preimages, equalityf−1(Y∖S)=f−1(Y)∖f−1(S){\displaystyle f^{-1}(Y\setminus S)=f^{-1}(Y)\setminus f^{-1}(S)}always holds.
The functionf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }defined byf(x)=x2{\displaystyle f(x)=x^{2}}is continuous, closed, and relatively open, but not (strongly) open. This is because ifU=(a,b){\displaystyle U=(a,b)}is any open interval inf{\displaystyle f}'s domainR{\displaystyle \mathbb {R} }that doesnotcontain0{\displaystyle 0}thenf(U)=(min{a2,b2},max{a2,b2}),{\displaystyle f(U)=(\min\{a^{2},b^{2}\},\max\{a^{2},b^{2}\}),}where this open interval is an open subset of bothR{\displaystyle \mathbb {R} }andImf:=f(R)=[0,∞).{\displaystyle \operatorname {Im} f:=f(\mathbb {R} )=[0,\infty ).}However, ifU=(a,b){\displaystyle U=(a,b)}is any open interval inR{\displaystyle \mathbb {R} }that contains0{\displaystyle 0}thenf(U)=[0,max{a2,b2}),{\displaystyle f(U)=[0,\max\{a^{2},b^{2}\}),}which is not an open subset off{\displaystyle f}'s codomainR{\displaystyle \mathbb {R} }butisan open subset ofImf=[0,∞).{\displaystyle \operatorname {Im} f=[0,\infty ).}Because the set of all open intervals inR{\displaystyle \mathbb {R} }is abasisfor theEuclidean topologyonR,{\displaystyle \mathbb {R} ,}this shows thatf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }is relatively open but not (strongly) open.
IfY{\displaystyle Y}has thediscrete topology(that is, all subsets are open and closed) then every functionf:X→Y{\displaystyle f:X\to Y}is both open and closed (but not necessarily continuous).
For example, thefloor functionfromR{\displaystyle \mathbb {R} }toZ{\displaystyle \mathbb {Z} }is open and closed, but not continuous.
This example shows that the image of aconnected spaceunder an open or closed map need not be connected.
Whenever we have aproductof topological spacesX=∏Xi,{\textstyle X=\prod X_{i},}the natural projectionspi:X→Xi{\displaystyle p_{i}:X\to X_{i}}are open[12][13](as well as continuous).
Since the projections offiber bundlesandcovering mapsare locally natural projections of products, these are also open maps.
Projections need not be closed however. Consider for instance the projectionp1:R2→R{\displaystyle p_{1}:\mathbb {R} ^{2}\to \mathbb {R} }on the first component; then the setA={(x,1/x):x≠0}{\displaystyle A=\{(x,1/x):x\neq 0\}}is closed inR2,{\displaystyle \mathbb {R} ^{2},}butp1(A)=R∖{0}{\displaystyle p_{1}(A)=\mathbb {R} \setminus \{0\}}is not closed inR.{\displaystyle \mathbb {R} .}However, for a compact spaceY,{\displaystyle Y,}the projectionX×Y→X{\displaystyle X\times Y\to X}is closed. This is essentially thetube lemma.
To every point on theunit circlewe can associate theangleof the positivex{\displaystyle x}-axis with the ray connecting the point with the origin. This function from the unit circle to the half-openinterval[0,2π) is bijective, open, and closed, but not continuous.
It shows that the image of acompact spaceunder an open or closed map need not be compact.
Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed. Specifying thecodomainis essential.
Everyhomeomorphismis open, closed, and continuous. In fact, abijectivecontinuous map is a homeomorphismif and only ifit is open, or equivalently, if and only if it is closed.
Thecompositionof two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map.[14][15]However, the composition of two relatively open maps need not be relatively open and similarly, the composition of two relatively closed maps need not be relatively closed.
Iff:X→Y{\displaystyle f:X\to Y}is strongly open (respectively, strongly closed) andg:Y→Z{\displaystyle g:Y\to Z}is relatively open (respectively, relatively closed) theng∘f:X→Z{\displaystyle g\circ f:X\to Z}is relatively open (respectively, relatively closed).
Letf:X→Y{\displaystyle f:X\to Y}be a map.
Given any subsetT⊆Y,{\displaystyle T\subseteq Y,}iff:X→Y{\displaystyle f:X\to Y}is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous,surjective) map then the same is true of its restrictionf|f−1(T):f−1(T)→T{\displaystyle f{\big \vert }_{f^{-1}(T)}~:~f^{-1}(T)\to T}to thef{\displaystyle f}-saturatedsubsetf−1(T).{\displaystyle f^{-1}(T).}
The categorical sum of two open maps is open, or of two closed maps is closed.[15]The categoricalproductof two open maps is open, however, the categorical product of two closed maps need not be closed.[14][15]
A bijective map is open if and only if it is closed.
The inverse of a bijective continuous map is a bijective open/closed map (and vice versa).
A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map. Alllocal homeomorphisms, including allcoordinate chartsonmanifoldsand allcovering maps, are open maps.
Closed map lemma—Every continuous functionf:X→Y{\displaystyle f:X\to Y}from acompact spaceX{\displaystyle X}to aHausdorff spaceY{\displaystyle Y}is closed andproper(meaning that preimages of compact sets are compact).
A variant of the closed map lemma states that if a continuous function betweenlocally compactHausdorff spaces is proper then it is also closed.
Incomplex analysis, the identically namedopen mapping theoremstates that every non-constantholomorphic functiondefined on aconnectedopen subset of thecomplex planeis an open map.
Theinvariance of domaintheorem states that a continuous and locally injective function between twon{\displaystyle n}-dimensionaltopological manifoldsmust be open.
Invariance of domain—IfU{\displaystyle U}is anopen subsetofRn{\displaystyle \mathbb {R} ^{n}}andf:U→Rn{\displaystyle f:U\to \mathbb {R} ^{n}}is aninjectivecontinuous map, thenV:=f(U){\displaystyle V:=f(U)}is open inRn{\displaystyle \mathbb {R} ^{n}}andf{\displaystyle f}is ahomeomorphismbetweenU{\displaystyle U}andV.{\displaystyle V.}
Infunctional analysis, theopen mapping theoremstates that every surjective continuouslinear operatorbetweenBanach spacesis an open map.
This theorem has been generalized totopological vector spacesbeyond just Banach spaces.
A surjective mapf:X→Y{\displaystyle f:X\to Y}is called analmost open mapif for everyy∈Y{\displaystyle y\in Y}there exists somex∈f−1(y){\displaystyle x\in f^{-1}(y)}such thatx{\displaystyle x}is apoint of opennessforf,{\displaystyle f,}which by definition means that for every open neighborhoodU{\displaystyle U}ofx,{\displaystyle x,}f(U){\displaystyle f(U)}is aneighborhoodoff(x){\displaystyle f(x)}inY{\displaystyle Y}(note that the neighborhoodf(U){\displaystyle f(U)}is not required to be anopenneighborhood).
Every surjective open map is an almost open map but in general, the converse is not necessarily true.
If a surjectionf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is an almost open map then it will be an open map if it satisfies the following condition (a condition that doesnotdepend in any way onY{\displaystyle Y}'s topologyσ{\displaystyle \sigma }):
If the map is continuous then the above condition is also necessary for the map to be open. That is, iff:X→Y{\displaystyle f:X\to Y}is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.
Iff:X→Y{\displaystyle f:X\to Y}is a continuous map that is also openorclosed then:
In the first two cases, being open or closed is merely asufficient conditionfor the conclusion that follows.
In the third case, it isnecessaryas well.
Iff:X→Y{\displaystyle f:X\to Y}is a continuous (strongly) open map,A⊆X,{\displaystyle A\subseteq X,}andS⊆Y,{\displaystyle S\subseteq Y,}then:
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Intopology, asubbase(orsubbasis,prebase,prebasis) for thetopologyτof atopological space(X, τ)is a subcollectionB{\displaystyle B}ofτ{\displaystyle \tau }that generatesτ,{\displaystyle \tau ,}in the sense thatτ{\displaystyle \tau }is the smallest topology containingB{\displaystyle B}as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.
Subbase is a weaker notion than that of abasefor a topology.
LetX{\displaystyle X}be a topological space with topologyτ.{\displaystyle \tau .}Asubbaseofτ{\displaystyle \tau }is usually defined as a subcollectionB{\displaystyle B}ofτ{\displaystyle \tau }satisfying one of the three following equivalent conditions:
If we additionally assume thatB{\displaystyle B}coversX{\displaystyle X}, or if we use thenullary intersectionconvention, then there is no need to includeX{\displaystyle X}in the third definition.
IfB{\displaystyle B}is a subbase ofτ{\displaystyle \tau }, we say thatB{\displaystyle B}generatesthe topologyτ.{\displaystyle \tau .}This terminology originates from the explicit construction ofτ{\displaystyle \tau }fromB{\displaystyle B}using the second or third definition above.
Elements of subbase are calledsubbasic (open) sets. Acovercomposed of subbasic sets is called asubbasic (open) cover.
ForanysubcollectionS{\displaystyle S}of thepower set℘(X),{\displaystyle \wp (X),}there is a unique topology havingS{\displaystyle S}as a subbase; it is the intersection of all topologies onX{\displaystyle X}containingS{\displaystyle S}. In general, however, the converse is not true, i.e. there is no unique subbasis for a given topology.
Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set℘(X){\displaystyle \wp (X)}and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the three conditions is more useful than the others.
Less commonly, a slightly different definition of subbase is given which requires that the subbaseB{\displaystyle {\mathcal {B}}}coverX.{\displaystyle X.}[2]In this case,X{\displaystyle X}is the union of all sets contained inB.{\displaystyle {\mathcal {B}}.}This means that there can be no confusion regarding the use of nullary intersections in the definition.
However, this definition is not always equivalent to the three definitions above. There exist topological spaces(X,τ){\displaystyle (X,\tau )}with subcollectionsB⊆τ{\displaystyle {\mathcal {B}}\subseteq \tau }of the topology such thatτ{\displaystyle \tau }is the smallest topology containingB{\displaystyle {\mathcal {B}}}, yetB{\displaystyle {\mathcal {B}}}does not coverX{\displaystyle X}. For example, consider a topological space(X,τ){\displaystyle (X,\tau )}withτ={∅,{p},X}{\displaystyle \tau =\{\varnothing ,\{p\},X\}}andB={{p}}{\displaystyle {\mathcal {B}}=\{\{p\}\}}for somep∈X.{\displaystyle p\in X.}Clearly,B{\displaystyle {\mathcal {B}}}is a subbase ofτ{\displaystyle \tau }, yetB{\displaystyle {\mathcal {B}}}doesn't coverX{\displaystyle X}as long asX{\displaystyle X}has at least2{\displaystyle 2}elements. In practice, this is a rare occurrence. E.g. a subbase of a space that has at least two points and satisfies theT1separation axiommust be a cover of that space.
The topology generated by any subsetS⊆{∅,X}{\displaystyle {\mathcal {S}}\subseteq \{\varnothing ,X\}}(including by the empty setS:=∅{\displaystyle {\mathcal {S}}:=\varnothing }) is equal to the trivial topology{∅,X}.{\displaystyle \{\varnothing ,X\}.}
Ifτ{\displaystyle \tau }is a topology onX{\displaystyle X}andB{\displaystyle {\mathcal {B}}}is a basis forτ{\displaystyle \tau }then the topology generated byB{\displaystyle {\mathcal {B}}}isτ.{\displaystyle \tau .}Thus any basisB{\displaystyle {\mathcal {B}}}for a topologyτ{\displaystyle \tau }is also a subbasis forτ.{\displaystyle \tau .}IfS{\displaystyle {\mathcal {S}}}is any subset ofτ{\displaystyle \tau }then the topology generated byS{\displaystyle {\mathcal {S}}}will be a subset ofτ.{\displaystyle \tau .}
The usual topology on thereal numbersR{\displaystyle \mathbb {R} }has a subbase consisting of allsemi-infiniteopen intervals either of the form(−∞,a){\displaystyle (-\infty ,a)}or(b,∞),{\displaystyle (b,\infty ),}wherea{\displaystyle a}andb{\displaystyle b}are real numbers. Together, these generate the usual topology, since the intersections(a,b)=(−∞,b)∩(a,∞){\displaystyle (a,b)=(-\infty ,b)\cap (a,\infty )}fora≤b{\displaystyle a\leq b}generate the usual topology. A second subbase is formed by taking the subfamily wherea{\displaystyle a}andb{\displaystyle b}arerational. The second subbase generates the usual topology as well, since the open intervals(a,b){\displaystyle (a,b)}witha,{\displaystyle a,}b{\displaystyle b}rational, are a basis for the usual Euclidean topology.
The subbase consisting of all semi-infinite open intervals of the form(−∞,a){\displaystyle (-\infty ,a)}alone, wherea{\displaystyle a}is a real number, does not generate the usual topology. The resulting topology does not satisfy theT1separation axiom, since ifa<b{\displaystyle a<b}everyopen setcontainingb{\displaystyle b}also containsa.{\displaystyle a.}
Theinitial topologyonX{\displaystyle X}defined by a family of functionsfi:X→Yi,{\displaystyle f_{i}:X\to Y_{i},}where eachYi{\displaystyle Y_{i}}has a topology, is the coarsest topology onX{\displaystyle X}such that eachfi{\displaystyle f_{i}}iscontinuous. Because continuity can be defined in terms of theinverse imagesof open sets, this means that the initial topology onX{\displaystyle X}is given by taking allfi−1(U),{\displaystyle f_{i}^{-1}(U),}whereU{\displaystyle U}ranges over all open subsets ofYi,{\displaystyle Y_{i},}as a subbasis.
Two important special cases of the initial topology are theproduct topology, where the family of functions is the set of projections from the product to each factor, and thesubspace topology, where the family consists of just one function, theinclusion map.
Thecompact-open topologyon the space of continuous functions fromX{\displaystyle X}toY{\displaystyle Y}has for a subbase the set of functionsV(K,U)={f:X→Y∣f(K)⊆U}{\displaystyle V(K,U)=\{f:X\to Y\mid f(K)\subseteq U\}}whereK⊆X{\displaystyle K\subseteq X}iscompactandU{\displaystyle U}is an open subset ofY.{\displaystyle Y.}
Suppose that(X,τ){\displaystyle (X,\tau )}is aHausdorfftopological space withX{\displaystyle X}containing two or more elements (for example,X=R{\displaystyle X=\mathbb {R} }with theEuclidean topology). LetY∈τ{\displaystyle Y\in \tau }be any non-emptyopensubset of(X,τ){\displaystyle (X,\tau )}(for example,Y{\displaystyle Y}could be a non-empty bounded open interval inR{\displaystyle \mathbb {R} }) and letν{\displaystyle \nu }denote thesubspace topologyonY{\displaystyle Y}thatY{\displaystyle Y}inherits from(X,τ){\displaystyle (X,\tau )}(soν⊆τ{\displaystyle \nu \subseteq \tau }). Then the topology generated byν{\displaystyle \nu }onX{\displaystyle X}is equal to the union{X}∪ν{\displaystyle \{X\}\cup \nu }(see the footnote for an explanation),[note 2]where{X}∪ν⊆τ{\displaystyle \{X\}\cup \nu \subseteq \tau }(since(X,τ){\displaystyle (X,\tau )}is Hausdorff, equality will hold if and only ifY=X{\displaystyle Y=X}). Note that ifY{\displaystyle Y}is aproper subsetofX,{\displaystyle X,}then{X}∪ν{\displaystyle \{X\}\cup \nu }is the smallest topologyonX{\displaystyle X}containingν{\displaystyle \nu }yetν{\displaystyle \nu }does not coverX{\displaystyle X}(that is, the union⋃V∈νV=Y{\displaystyle \bigcup _{V\in \nu }V=Y}is a proper subset ofX{\displaystyle X}).
One nice fact about subbases is thatcontinuityof a function need only be checked on a subbase of the range. That is, iff:X→Y{\displaystyle f:X\to Y}is a map between topological spaces and ifB{\displaystyle {\mathcal {B}}}is a subbase forY,{\displaystyle Y,}thenf:X→Y{\displaystyle f:X\to Y}is continuousif and only iff−1(B){\displaystyle f^{-1}(B)}is open inX{\displaystyle X}for everyB∈B.{\displaystyle B\in {\mathcal {B}}.}Anet(orsequence)x∙=(xi)i∈I{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}converges to a pointx{\displaystyle x}if and only if everysubbasic neighborhood ofx{\displaystyle x}contains allxi{\displaystyle x_{i}}for sufficiently largei∈I.{\displaystyle i\in I.}
The Alexander Subbase Theorem is a significant result concerning subbases that is due toJames Waddell Alexander II.[3]The corresponding result for basic (rather than subbasic) open covers is much easier to prove.
The converse to this theorem also holds (because every cover ofX{\displaystyle X}by elements ofS{\displaystyle {\mathcal {S}}}is an open cover ofX{\displaystyle X})
Suppose for the sake of contradiction that the spaceX{\displaystyle X}is not compact (soX{\displaystyle X}is an infinite set), yet every subbasic cover fromS{\displaystyle {\mathcal {S}}}has a finite subcover.
LetS{\displaystyle \mathbb {S} }denote the set of all open covers ofX{\displaystyle X}that do not have any finite subcover ofX.{\displaystyle X.}Partially orderS{\displaystyle \mathbb {S} }by subset inclusion and useZorn's Lemmato find an elementC∈S{\displaystyle {\mathcal {C}}\in \mathbb {S} }that is a maximal element ofS.{\displaystyle \mathbb {S} .}Observe that:
We will begin by showing thatC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}isnota cover ofX.{\displaystyle X.}Suppose thatC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}was a cover ofX,{\displaystyle X,}which in particular implies thatC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}is a cover ofX{\displaystyle X}by elements ofS.{\displaystyle {\mathcal {S}}.}The theorem's hypothesis onS{\displaystyle {\mathcal {S}}}implies that there exists a finite subset ofC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}that coversX,{\displaystyle X,}which would simultaneously also be a finite subcover ofX{\displaystyle X}by elements ofC{\displaystyle {\mathcal {C}}}(sinceC∩S⊆C{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}\subseteq {\mathcal {C}}}).
But this contradictsC∈S,{\displaystyle {\mathcal {C}}\in \mathbb {S} ,}which proves thatC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}does not coverX.{\displaystyle X.}
SinceC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}does not coverX,{\displaystyle X,}there exists somex∈X{\displaystyle x\in X}that is not covered byC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}(that is,x{\displaystyle x}is not contained in any element ofC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}).
But sinceC{\displaystyle {\mathcal {C}}}does coverX,{\displaystyle X,}there also exists someU∈C{\displaystyle U\in {\mathcal {C}}}such thatx∈U.{\displaystyle x\in U.}It follows thatU≠X{\displaystyle U\neq X}, because otherwise it would implyC{\displaystyle {\mathcal {C}}}has a finite subcover ofX{\displaystyle X}, namely the subcover{U}={X},{\displaystyle \{U\}=\{X\},}contradictingC∈S.{\displaystyle {\mathcal {C}}\in \mathbb {S} .}SinceU≠X,{\displaystyle U\neq X,}andS{\displaystyle {\mathcal {S}}}is a subbasis generatingX{\displaystyle X}'s topology (together withX{\displaystyle X}), from the definition of the topology generated byS,{\displaystyle {\mathcal {S}},}there must exist a finite collection of subbasic open setsS1,…,Sn∈S{\displaystyle S_{1},\ldots ,S_{n}\in {\mathcal {S}}}withn≥1{\displaystyle n\geq 1}such thatx∈S1∩⋯∩Sn⊆U.{\displaystyle x\in S_{1}\cap \cdots \cap S_{n}\subseteq U.}
We will now show by contradiction thatSi∉C{\displaystyle S_{i}\not \in {\mathcal {C}}}for everyi=1,…,n.{\displaystyle i=1,\ldots ,n.}Ifi{\displaystyle i}was such thatSi∈C,{\displaystyle S_{i}\in {\mathcal {C}},}then alsoSi∈C∩S{\displaystyle S_{i}\in {\mathcal {C}}\cap {\mathcal {S}}}so the fact thatx∈Si{\displaystyle x\in S_{i}}would then imply thatx{\displaystyle x}is covered byC∩S,{\displaystyle {\mathcal {C}}\cap {\mathcal {S}},}which contradicts howx{\displaystyle x}was chosen (recall thatx{\displaystyle x}was chosen specifically so that it was not covered byC∩S{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}).
As mentioned earlier, the maximality ofC{\displaystyle {\mathcal {C}}}inS{\displaystyle \mathbb {S} }implies that for everyi=1,…,n,{\displaystyle i=1,\ldots ,n,}there exists a finite subsetCSi{\displaystyle {\mathcal {C}}_{S_{i}}}ofC{\displaystyle {\mathcal {C}}}such that{Si}∪CSi{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{S_{i}}}forms a finite cover ofX.{\displaystyle X.}DefineCF:=CS1∪⋯∪CSn,{\displaystyle {\mathcal {C}}_{F}:={\mathcal {C}}_{S_{1}}\cup \cdots \cup {\mathcal {C}}_{S_{n}},}which is a finite subset ofC.{\displaystyle {\mathcal {C}}.}Observe that for everyi=1,…,n,{\displaystyle i=1,\ldots ,n,}{Si}∪CF{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}is a finite cover ofX{\displaystyle X}so let us replace everyCSi{\displaystyle {\mathcal {C}}_{S_{i}}}withCF.{\displaystyle {\mathcal {C}}_{F}.}
Let∪CF{\displaystyle \cup {\mathcal {C}}_{F}}denote the union of all sets inCF{\displaystyle {\mathcal {C}}_{F}}(which is an open subset ofX{\displaystyle X}) and letZ{\displaystyle Z}denote the complement of∪CF{\displaystyle \cup {\mathcal {C}}_{F}}inX.{\displaystyle X.}Observe that for any subsetA⊆X,{\displaystyle A\subseteq X,}{A}∪CF{\displaystyle \{A\}\cup {\mathcal {C}}_{F}}coversX{\displaystyle X}if and only ifZ⊆A.{\displaystyle Z\subseteq A.}In particular, for everyi=1,…,n,{\displaystyle i=1,\ldots ,n,}the fact that{Si}∪CF{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}coversX{\displaystyle X}implies thatZ⊆Si.{\displaystyle Z\subseteq S_{i}.}Sincei{\displaystyle i}was arbitrary, we haveZ⊆S1∩⋯∩Sn.{\displaystyle Z\subseteq S_{1}\cap \cdots \cap S_{n}.}Recalling thatS1∩⋯∩Sn⊆U,{\displaystyle S_{1}\cap \cdots \cap S_{n}\subseteq U,}we thus haveZ⊆U,{\displaystyle Z\subseteq U,}which is equivalent to{U}∪CF{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}being a cover ofX.{\displaystyle X.}Moreover,{U}∪CF{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}is a finite cover ofX{\displaystyle X}with{U}∪CF⊆C.{\displaystyle \{U\}\cup {\mathcal {C}}_{F}\subseteq {\mathcal {C}}.}ThusC{\displaystyle {\mathcal {C}}}has a finite subcover ofX,{\displaystyle X,}which contradicts the fact thatC∈S.{\displaystyle {\mathcal {C}}\in \mathbb {S} .}Therefore, the original assumption thatX{\displaystyle X}is not compact must be wrong, which proves thatX{\displaystyle X}is compact.◼{\displaystyle \blacksquare }
Although this proof makes use ofZorn's Lemma, the proof does not need the full strength of choice.
Instead, it relies on the intermediateUltrafilter principle.[3]
Using this theorem with the subbase forR{\displaystyle \mathbb {R} }above, one can give a very easy proof that bounded closed intervals inR{\displaystyle \mathbb {R} }are compact.
More generally,Tychonoff's theorem, which states that the product of non-empty compact spaces is compact, has a short proof if the Alexander Subbase Theorem is used.
The product topology on∏iXi{\displaystyle \prod _{i}X_{i}}has, by definition, a subbase consisting ofcylindersets that are the inverse projections of an open set in one factor.
Given asubbasicfamilyC{\displaystyle C}of the product that does not have a finite subcover, we can partitionC=∪iCi{\displaystyle C=\cup _{i}C_{i}}into subfamilies that consist of exactly those cylinder sets corresponding to a given factor space.
By assumption, ifCi≠∅{\displaystyle C_{i}\neq \varnothing }thenCi{\displaystyle C_{i}}doesnothave a finite subcover.
Being cylinder sets, this means their projections ontoXi{\displaystyle X_{i}}have no finite subcover, and since eachXi{\displaystyle X_{i}}is compact, we can find a pointxi∈Xi{\displaystyle x_{i}\in X_{i}}that is not covered by the projections ofCi{\displaystyle C_{i}}ontoXi.{\displaystyle X_{i}.}But then(xi)i∈∏iXi{\displaystyle \left(x_{i}\right)_{i}\in \prod _{i}X_{i}}is not covered byC.{\displaystyle C.}◼{\displaystyle \blacksquare }
Note, that in the last step we implicitly used theaxiom of choice(which is actually equivalent toZorn's lemma) to ensure the existence of(xi)i.{\displaystyle \left(x_{i}\right)_{i}.}
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https://en.wikipedia.org/wiki/Subbase
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Inmathematics, apointxis called anisolated pointof a subsetS(in atopological spaceX) ifxis an element ofSand there exists aneighborhoodofxthat does not contain any other points ofS. This is equivalent to saying that thesingleton{x}is anopen setin the topological spaceS(considered as asubspaceofX). Another equivalent formulation is: an elementxofSis an isolated point ofSif and only if it is not alimit pointofS.
If the spaceXis ametric space, for example aEuclidean space, then an elementxofSis an isolated point ofSif there exists anopen ballaroundxthat contains only finitely many elements ofS.
Apoint setthat is made up only of isolated points is called adiscrete setordiscrete point set(see alsodiscrete space).
Any discrete subsetSof Euclidean space must becountable, since the isolation of each of its points together with the fact thatrationalsaredensein therealsmeans that the points ofSmay be mapped injectively onto a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example.
A set with no isolated point is said to bedense-in-itself(every neighbourhood of a point contains other points of the set). Aclosed setwith no isolated point is called aperfect set(it contains all its limit points and no isolated points).
The number of isolated points is atopological invariant, i.e. if twotopological spacesX, Yarehomeomorphic, the number of isolated points in each is equal.
Topological spacesin the following three examples are considered assubspacesof thereal linewith the standard topology.
In the topological spaceX={a,b}{\displaystyle X=\{a,b\}}with topologyτ={∅,{a},X},{\displaystyle \tau =\{\emptyset ,\{a\},X\},}the elementais an isolated point, even thoughb{\displaystyle b}belongs to theclosureof{a}{\displaystyle \{a\}}(and is therefore, in some sense, "close" toa). Such a situation is not possible in aHausdorff space.
TheMorse lemmastates thatnon-degenerate critical pointsof certain functions are isolated.
Consider the setFof pointsxin the real interval(0,1)such that every digitxiof theirbinaryrepresentation fulfills the following conditions:
Informally, these conditions means that every digit of the binary representation ofx{\displaystyle x}that equals 1 belongs to a pair ...0110..., except for ...010... at the very end.
Now,Fis an explicit set consisting entirely of isolated points but has the counter-intuitive property that itsclosureis anuncountable set.[1]
Another setFwith the same properties can be obtained as follows. LetCbe the middle-thirdsCantor set, letI1,I2,I3,…,Ik,…{\displaystyle I_{1},I_{2},I_{3},\ldots ,I_{k},\ldots }be thecomponentintervals of[0,1]−C{\displaystyle [0,1]-C}, and letFbe a set consisting of one point from eachIk. Since eachIkcontains only one point fromF, every point ofFis an isolated point. However, ifpis any point in the Cantor set, then every neighborhood ofpcontains at least oneIk, and hence at least one point ofF. It follows that each point of the Cantor set lies in the closure ofF, and thereforeFhas uncountable closure.
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https://en.wikipedia.org/wiki/Isolated_point
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Intopologyand related areas ofmathematics, theneighbourhood system,complete system of neighbourhoods,[1]orneighbourhood filterN(x){\displaystyle {\mathcal {N}}(x)}for a pointx{\displaystyle x}in atopological spaceis the collection of allneighbourhoodsofx.{\displaystyle x.}
Neighbourhood of a point or set
Anopen neighbourhoodof a point (orsubset[note 1])x{\displaystyle x}in a topological spaceX{\displaystyle X}is anyopen subsetU{\displaystyle U}ofX{\displaystyle X}that containsx.{\displaystyle x.}Aneighbourhoodofx{\displaystyle x}inX{\displaystyle X}is any subsetN⊆X{\displaystyle N\subseteq X}that containssomeopen neighbourhood ofx{\displaystyle x};
explicitly,N{\displaystyle N}is a neighbourhood ofx{\displaystyle x}inX{\displaystyle X}if and only ifthere exists some open subsetU{\displaystyle U}withx∈U⊆N{\displaystyle x\in U\subseteq N}.[2][3]Equivalently, a neighborhood ofx{\displaystyle x}is any set that containsx{\displaystyle x}in itstopological interior.
Importantly, a "neighbourhood" doesnothave to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods."[note 2]Similarly, a neighbourhood that is also aclosed(respectively,compact,connected, etc.) set is called aclosed neighbourhood(respectively,compact neighbourhood,connected neighbourhood, etc.).
There are many other types of neighbourhoods that are used in topology and related fields likefunctional analysis.
The family of all neighbourhoods having a certain "useful" property often forms aneighbourhood basis, although many times, these neighbourhoods are not necessarily open.Locally compact spaces, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.
Neighbourhood filter
The neighbourhood system for a point (ornon-emptysubset)x{\displaystyle x}is afiltercalled theneighbourhood filterforx.{\displaystyle x.}The neighbourhood filter for a pointx∈X{\displaystyle x\in X}is the same as the neighbourhood filter of thesingleton set{x}.{\displaystyle \{x\}.}
Aneighbourhood basisorlocal basis(orneighbourhood baseorlocal base) for a pointx{\displaystyle x}is afilter baseof the neighbourhood filter; this means that it is a subsetB⊆N(x){\displaystyle {\mathcal {B}}\subseteq {\mathcal {N}}(x)}such that for allV∈N(x),{\displaystyle V\in {\mathcal {N}}(x),}there exists someB∈B{\displaystyle B\in {\mathcal {B}}}such thatB⊆V.{\displaystyle B\subseteq V.}[3]Here,N(x){\displaystyle {\mathcal {N}}(x)}denotes the set of all neighbourhoods of x.
That is, for any neighbourhoodV{\displaystyle V}we can find a neighbourhoodB{\displaystyle B}in the neighbourhood basis that is contained inV.{\displaystyle V.}
Equivalently,B{\displaystyle {\mathcal {B}}}is a local basis atx{\displaystyle x}if and only if the neighbourhood filterN{\displaystyle {\mathcal {N}}}can be recovered fromB{\displaystyle {\mathcal {B}}}in the sense that the following equality holds:[4]N(x)={V⊆X:B⊆Vfor someB∈B}.{\displaystyle {\mathcal {N}}(x)=\left\{V\subseteq X~:~B\subseteq V{\text{ for some }}B\in {\mathcal {B}}\right\}\!\!\;.}A familyB⊆N(x){\displaystyle {\mathcal {B}}\subseteq {\mathcal {N}}(x)}is a neighbourhood basis forx{\displaystyle x}if and only ifB{\displaystyle {\mathcal {B}}}is acofinal subsetof(N(x),⊇){\displaystyle \left({\mathcal {N}}(x),\supseteq \right)}with respect to thepartial order⊇{\displaystyle \supseteq }(importantly, this partial order is thesupersetrelation and not thesubsetrelation).
Aneighbourhood subbasisatx{\displaystyle x}is a familyS{\displaystyle {\mathcal {S}}}of subsets ofX,{\displaystyle X,}each of which containsx,{\displaystyle x,}such that the collection of all possible finiteintersectionsof elements ofS{\displaystyle {\mathcal {S}}}forms a neighbourhood basis atx.{\displaystyle x.}
IfR{\displaystyle \mathbb {R} }has its usualEuclidean topologythen the neighborhoods of0{\displaystyle 0}are all those subsetsN⊆R{\displaystyle N\subseteq \mathbb {R} }for which there exists somereal numberr>0{\displaystyle r>0}such that(−r,r)⊆N.{\displaystyle (-r,r)\subseteq N.}For example, all of the following sets are neighborhoods of0{\displaystyle 0}inR{\displaystyle \mathbb {R} }:(−2,2),[−2,2],[−2,∞),[−2,2)∪{10},[−2,2]∪Q,R{\displaystyle (-2,2),\;[-2,2],\;[-2,\infty ),\;[-2,2)\cup \{10\},\;[-2,2]\cup \mathbb {Q} ,\;\mathbb {R} }but none of the following sets are neighborhoods of0{\displaystyle 0}:{0},Q,(0,2),[0,2),[0,2)∪Q,(−2,2)∖{1,12,13,14,…}{\displaystyle \{0\},\;\mathbb {Q} ,\;(0,2),\;[0,2),\;[0,2)\cup \mathbb {Q} ,\;(-2,2)\setminus \left\{1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\ldots \right\}}whereQ{\displaystyle \mathbb {Q} }denotes therational numbers.
IfU{\displaystyle U}is an open subset of atopological spaceX{\displaystyle X}then for everyu∈U,{\displaystyle u\in U,}U{\displaystyle U}is a neighborhood ofu{\displaystyle u}inX.{\displaystyle X.}More generally, ifN⊆X{\displaystyle N\subseteq X}is any set andintXN{\displaystyle \operatorname {int} _{X}N}denotes thetopological interiorofN{\displaystyle N}inX,{\displaystyle X,}thenN{\displaystyle N}is a neighborhood (inX{\displaystyle X}) of every pointx∈intXN{\displaystyle x\in \operatorname {int} _{X}N}and moreover,N{\displaystyle N}isnota neighborhood of any other point.
Said differently,N{\displaystyle N}is a neighborhood of a pointx∈X{\displaystyle x\in X}if and only ifx∈intXN.{\displaystyle x\in \operatorname {int} _{X}N.}
Neighbourhood bases
In any topological space, the neighbourhood system for a point is also a neighbourhood basis for the point. The set of all open neighbourhoods at a point forms a neighbourhood basis at that point.
For any pointx{\displaystyle x}in ametric space, the sequence ofopen ballsaroundx{\displaystyle x}with radius1/n{\displaystyle 1/n}form acountableneighbourhood basisB={B1/n:n=1,2,3,…}{\displaystyle {\mathcal {B}}=\left\{B_{1/n}:n=1,2,3,\dots \right\}}. This means every metric space isfirst-countable.
Given a spaceX{\displaystyle X}with theindiscrete topologythe neighbourhood system for any pointx{\displaystyle x}only contains the whole space,N(x)={X}{\displaystyle {\mathcal {N}}(x)=\{X\}}.
In theweak topologyon the space of measures on a spaceE,{\displaystyle E,}a neighbourhood base aboutν{\displaystyle \nu }is given by{μ∈M(E):|μfi−νfi|<ri,i=1,…,n}{\displaystyle \left\{\mu \in {\mathcal {M}}(E):\left|\mu f_{i}-\nu f_{i}\right|<r_{i},\,i=1,\dots ,n\right\}}wherefi{\displaystyle f_{i}}arecontinuousbounded functions fromE{\displaystyle E}to the real numbers andr1,…,rn{\displaystyle r_{1},\dots ,r_{n}}are positive real numbers.
Seminormed spaces and topological groups
In aseminormed space, that is avector spacewith thetopologyinduced by aseminorm, all neighbourhood systems can be constructed bytranslationof the neighbourhood system for the origin,N(x)=N(0)+x.{\displaystyle {\mathcal {N}}(x)={\mathcal {N}}(0)+x.}
This is because, by assumption, vector addition is separately continuous in the induced topology. Therefore, the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the space is atopological groupor the topology is defined by apseudometric.
Supposeu∈U⊆X{\displaystyle u\in U\subseteq X}and letN{\displaystyle {\mathcal {N}}}be a neighbourhood basis foru{\displaystyle u}inX.{\displaystyle X.}MakeN{\displaystyle {\mathcal {N}}}into adirected setbypartially orderingit by superset inclusion⊇.{\displaystyle \,\supseteq .}ThenU{\displaystyle U}isnota neighborhood ofu{\displaystyle u}inX{\displaystyle X}if and only if there exists anN{\displaystyle {\mathcal {N}}}-indexednet(xN)N∈N{\displaystyle \left(x_{N}\right)_{N\in {\mathcal {N}}}}inX∖U{\displaystyle X\setminus U}such thatxN∈N∖U{\displaystyle x_{N}\in N\setminus U}for everyN∈N{\displaystyle N\in {\mathcal {N}}}(which implies that(xN)N∈N→u{\displaystyle \left(x_{N}\right)_{N\in {\mathcal {N}}}\to u}inX{\displaystyle X}).
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https://en.wikipedia.org/wiki/Neighbourhood_system
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