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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can b... | Wikipedia/Boolean_algebra_(structure) |
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domai... | Wikipedia/Domain_(ring_theory) |
Let
ϕ
:
M
→
N
{\displaystyle \phi :M\to N}
be a smooth map between smooth manifolds
M
{\displaystyle M}
and
N
{\displaystyle N}
. Then there is an associated... | Wikipedia/Pullback_(differential_geometry) |
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The rep... | Wikipedia/Hopf_algebra |
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra.: 46 The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homo... | Wikipedia/Bialgebra |
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space
C
n
... | Wikipedia/Holomorphic_function |
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that
v
u
=
u
v
=
1
,
{\displaystyle vu=u... | Wikipedia/Unit_(algebra) |
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that
φ
:
M
→
N
{\displaystyle \varphi \colon M\to N}
is a smooth map between smooth manifolds; then the differentia... | Wikipedia/Pushforward_(differential) |
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b called implication such that (c ∧ a) ≤ b is equivalent to c ≤ (a → b). From a logical stand... | Wikipedia/Heyting_algebra |
In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring R, with an operad replacing R.
== Definitions ==
Given an operad O (say, a symmetric sequence in a symmetric monoidal ∞-category C), an algebra over an operad, or O-algebra for short,... | Wikipedia/Algebra_over_an_operad |
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be a... | Wikipedia/Continuous_function |
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
A ... | Wikipedia/B*-algebra |
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
x
(
x
y
)
=
(
x
x
)
y
{\displaystyle x(xy)=(xx)y}
... | Wikipedia/Alternative_algebra |
In mathematics, the composition operator
takes two functions,
f
{\displaystyle f}
and
g
{\displaystyle g}
, and returns a new function
h
(
x
)
:=
(
g
∘... | Wikipedia/Functional_composition |
In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity:
a
∙
(
b
∙
a
)
=
(
... | Wikipedia/Flexible_algebra |
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras a... | Wikipedia/Differential_algebra |
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and nu... | Wikipedia/Incidence_algebra |
In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original.
== Definitions ==
Let A be an algebra over a field F with multiplication (... | Wikipedia/Mutation_(algebra) |
In algebra, Zariski's lemma, proved by Oscar Zariski (1947), states that, if a field K is finitely generated as an associative algebra over another field k, then K is a finite field extension of k (that is, it is also finitely generated as a vector space).
An important application of the lemma is a proof of the weak fo... | Wikipedia/Zariski's_lemma |
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques... | Wikipedia/Operator_algebra |
In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively.
The tropical semiring has various applications (see tropical analysis), and forms the b... | Wikipedia/Max-plus_algebra |
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
== Definition ==
An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that... | Wikipedia/Power-associative_algebra |
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x1 ≠ x2 implies f(x1) ≠ f(x2) (equivalently by contraposition, f(x1) = f(x2) implies x1 = x2). In other words, every element ... | Wikipedia/Injective_function |
In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.
For every vector space there exists a basis, and all ... | Wikipedia/Dimension_(linear_algebra) |
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
A ... | Wikipedia/C*-algebra |
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a real-valued function... | Wikipedia/Newton's_method |
In mathematics and, more specifically, in theory of equations, the principal form of an irreducible polynomial of degree at least three is a polynomial of the same degree n without terms of degrees n−1 and n−2, such that each root of either polynomial is a rational function of a root of the other polynomial.
The princi... | Wikipedia/Principal_equation_form |
In numerical analysis, the Weierstrass method or Durand–Kerner method, discovered by Karl Weierstrass in 1891 and rediscovered independently by Durand in 1960 and Kerner in 1966, is a root-finding algorithm for solving polynomial equations. In other words, the method can be used to solve numerically the equation f(x)=0... | Wikipedia/Durand–Kerner_method |
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f
(
z
)
=
a
z
+
b
c
... | Wikipedia/Möbius_transformation |
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case.
The definitions of many classes of composite or structured objects often implicitly incl... | Wikipedia/Degenerate_case |
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or takin... | Wikipedia/Noncommutative_algebraic_geometry |
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
== Group theory ==
The commutator of two elements, g and h, of a group G, is the element
[g, h] = g−1h−1gh.
This element ... | Wikipedia/Commutator_(ring_theory) |
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later ... | Wikipedia/Zariski_topology |
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under ... | Wikipedia/Invariant_theory |
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise se... | Wikipedia/Nakayama's_lemma |
Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. It is on the list of 173 books essential for undergraduate math libraries.
As of May 2025, Google Scholar lists over 8000 citations to this book.
It deals with elementary concepts of commuta... | Wikipedia/Introduction_to_Commutative_Algebra |
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions
m
s
... | Wikipedia/Localization_(algebra) |
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Alexander Grothendieck to define étal... | Wikipedia/Étale_topology |
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.)
Throughout the articles, rings refer to... | Wikipedia/Ideal_theory |
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. More generally, if a polynomial... | Wikipedia/Hensel's_lemma |
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure th... | Wikipedia/Completion_(ring_theory) |
Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedr... | Wikipedia/Combinatorial_commutative_algebra |
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... | Wikipedia/Duality_(category_theory) |
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image.
When the algebraic structures involved have an underlying group struc... | Wikipedia/Kernel_(algebra) |
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ... | Wikipedia/Scheme_theory |
In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are
3
x
x
... | Wikipedia/Algebraic_fraction |
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of ... | Wikipedia/Commutative_algebra_(structure) |
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, s... | Wikipedia/Algebraic_integer |
In mathematics, an algebraic number field (or simply number field) is an extension field
K
{\displaystyle K}
of the field of rational numbers
Q
{\displaystyle \mathbb {Q} }
such that the field extension
... | Wikipedia/Algebraic_number_field |
In algebra, the free product (coproduct) of a family of associative algebras
A
i
,
i
∈
I
{\displaystyle A_{i},i\in I}
over a commutative ring R is the associative algebra over R that is, rou... | Wikipedia/Free_product_of_associative_algebras |
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations.
== Definition ==
Let R be a commutative ring and l... | Wikipedia/Tensor_product_of_algebras |
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.
More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial rin... | Wikipedia/Affine_algebraic_variety |
In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S.
Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero a... | Wikipedia/Annihilator_(ring_theory) |
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions
m
s
... | Wikipedia/Localization_(commutative_algebra) |
In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the ... | Wikipedia/Nisnevich_topology |
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.
Grothendieck topologies axiomatize the notion of an open cover. Us... | Wikipedia/Grothendieck_topology |
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (eve... | Wikipedia/Finitely_generated_ideal |
In mathematics, a base (or basis; pl.: bases) for the topology τ of a topological space (X, τ) is a family
B
{\displaystyle {\mathcal {B}}}
of open subsets of X such that every open set of the topology is equal to the union of some ... | Wikipedia/Basis_(topology) |
Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline (or offshoot) of theoretical p... | Wikipedia/Computational_physics |
Theoretical computer science is a subfield of computer science and mathematics that focuses on the abstract and mathematical foundations of computation.
It is difficult to circumscribe the theoretical areas precisely. The ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following ... | Wikipedia/Theoretical_computer_science |
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can de... | Wikipedia/Von_Neumann–Bernays–Gödel_set_theory |
A graphics processing unit (GPU) is a specialized electronic circuit designed for digital image processing and to accelerate computer graphics, being present either as a discrete video card or embedded on motherboards, mobile phones, personal computers, workstations, and game consoles. GPUs were later found to be usefu... | Wikipedia/Graphics_processing_unit |
In mathematics, a surjective function (also known as surjection, or onto function ) is a function f such that, for every element y of the function's codomain, there exists at least one element x in the function's domain such that f(x) = y. In other words, for a function f : X → Y, the codomain Y is the image of the fun... | Wikipedia/Surjective_function |
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by ... | Wikipedia/Randomized_algorithm |
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary properties of 0, 1, +, ×,
x
y
... | Wikipedia/Elementary_function_arithmetic |
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory (MLTT)) is a type theory and an alternative foundation of mathematics.
Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versi... | Wikipedia/Martin-Löf_type_theory |
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory (MLTT)) is a type theory and an alternative foundation of mathematics.
Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versi... | Wikipedia/Intuitionistic_type_theory |
A typed lambda calculus is a typed formalism that uses the lambda symbol (
λ
{\displaystyle \lambda }
) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depen... | Wikipedia/Typed_lambda_calculus |
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, giving rise to a formal system that combines the language with deduction rules. An element
ϕ
∈
T
... | Wikipedia/Theory_(mathematical_logic) |
In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N. For example, every reduct or definitional expansion of a structure N has an interpretation in N.
Many model-theoretic properties are p... | Wikipedia/Interpretation_(model_theory) |
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x1, x2,..., xn that are true of a set ... | Wikipedia/Type_(model_theory) |
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can b... | Wikipedia/Axiomatization_of_Boolean_algebras |
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computab... | Wikipedia/Computable_function |
In mathematical logic, an uninterpreted function or function symbol is one that has no other property than its name and n-ary form. Function symbols are used, together with constants and variables, to form terms.
The theory of uninterpreted functions is also sometimes called the free theory, because it is freely genera... | Wikipedia/Uninterpreted_function |
Visualization (or visualisation ), also known as graphics visualization, is any technique for creating images, diagrams, or animations to communicate a message. Visualization through visual imagery has been an effective way to communicate both abstract and concrete ideas since the dawn of humanity. from history include... | Wikipedia/Visualization_(graphics) |
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it should not be confused with first-order logic. It ... | Wikipedia/Propositional_calculus |
In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term.
Functional predicates are also sometimes called mappings, but that term has additional meanings in mathematics.
In a model, a functio... | Wikipedia/Functional_predicate |
An integrated circuit (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components are etched onto a small, flat piece ("chip") of semiconductor material, usually... | Wikipedia/Integrated_circuit |
In mathematical logic, a theory can be extended with
new constants or function names under certain conditions with assurance that the extension will introduce
no contradiction. Extension by definitions is perhaps the best-known approach, but it requires
unique existence of an object with the desired property. Addition... | Wikipedia/Extension_by_new_constant_and_function_names |
A dialogue system, or conversational agent (CA), is a computer system intended to converse with a human. Dialogue systems employed one or more of text, speech, graphics, haptics, gestures, and other modes for communication on both the input and output channel.
The elements of a dialogue system are not defined because t... | Wikipedia/Dialogue_systems |
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the size of an algorithm's input to the number of steps it takes (its ti... | Wikipedia/Analysis_of_algorithms |
In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assi... | Wikipedia/Calculus_of_constructions |
Type theory with records is a formal semantics representation framework, using records to express type theory types. It has been used in natural language processing, principally computational semantics and dialogue systems.
== Syntax ==
A record type is a set of fields. A field is a pair consisting of a label and a t... | Wikipedia/Type_theory_with_records |
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal... | Wikipedia/Sequent_calculus |
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop is fixed before entering the loop). Primitive recursive functions form a strict subset of... | Wikipedia/Primitive_recursive_function |
Cryptography, or cryptology (from Ancient Greek: κρυπτός, romanized: kryptós "hidden, secret"; and γράφειν graphein, "to write", or -λογία -logia, "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about const... | Wikipedia/Cryptography |
Form factor is a hardware design aspect that defines and prescribes the size, shape, and other physical specifications of components, particularly in electronics. A form factor may represent a broad class of similarly sized components, or it may prescribe a specific standard. It may also define an entire system, as in ... | Wikipedia/Form_factor_(design) |
In computer science, algorithmic efficiency is a property of an algorithm which relates to the amount of computational resources used by the algorithm. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or continuous process.
For maximum efficiency it is desirable to minim... | Wikipedia/Algorithmic_efficiency |
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.
== Ontology ==
The ontology of GST is identical to that of ZFC, and henc... | Wikipedia/General_set_theory |
Interaction design, often abbreviated as IxD, is "the practice of designing interactive digital products, environments, systems, and services.": xxvii, 30 While interaction design has an interest in form (similar to other design fields), its main area of focus rests on behavior.: xxvii, 30 Rather than analyzing how t... | Wikipedia/Interaction_design |
An integrated development environment (IDE) is a software application that provides comprehensive facilities for software development. An IDE normally consists of at least a source-code editor, build automation tools, and a debugger. Some IDEs, such as IntelliJ IDEA, Eclipse and Lazarus contain the necessary compiler, ... | Wikipedia/Integrated_development_environment |
Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas:
Theoretical foundations and analysis
Use of computer technology to aid logicians
Use of concepts from logic for computer applications
== Theoretical founda... | Wikipedia/Logic_in_computer_science |
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays... | Wikipedia/Morse–Kelley_set_theory |
In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.
This includes, among other lines of work, the construction of homoto... | Wikipedia/Homotopy_type_theory |
Network security is a umbrella term to describe security controls, policies, processes and practices adopted to prevent, detect and monitor unauthorized access, misuse, modification, or denial of a computer network and network-accessible resources. Network security involves the authorization of access to data in a netw... | Wikipedia/Network_security |
In mathematics and logic, Ackermann set theory (AST, also known as
A
∗
/
V
{\displaystyle A^{*}/V}
) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.
AST differs from Zer... | Wikipedia/Ackermann_set_theory |
In mathematical logic, and particularly in its subfield model theory, a saturated model M is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is
ℵ
1
... | Wikipedia/Saturated_model |
In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial se... | Wikipedia/Non-standard_model_of_arithmetic |
In model theory, a branch of mathematical logic, the spectrum of a theory
is given by the number of isomorphism classes of models in various cardinalities. More precisely,
for any complete theory T in a language we write I(T, κ) for the number of models of T (up to isomorphism) of cardinality κ. The spectrum problem ... | Wikipedia/Spectrum_of_a_theory |
In computer science, control flow (or flow of control) is the order in which individual statements, instructions or function calls of an imperative program are executed or evaluated. The emphasis on explicit control flow distinguishes an imperative programming language from a declarative programming language.
Within an... | Wikipedia/Control_flow |
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