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In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f: A → B {\displaystyle f:A\to B} . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups ⋯ → π n + 1 ( B ) → π n ( Hofiber ( f ) ) → π n ( A ) → π n ( B ) → ⋯ {\displaystyle \cdots \to \pi _{n+1}(B)\to \pi _{n}({\text{Hofiber}}(f))\to \pi _{n}(A)\to \pi _{n}(B)\to \cdots } Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle C ( f ) ∙ → A ∙ → B ∙ → {\displaystyle C(f)_{\bullet }\to A_{\bullet }\to B_{\bullet }\xrightarrow {} } gives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber.
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https://en.wikipedia.org/wiki/Homotopy_fiber
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In mathematics, especially homotopy theory, the mapping cone is a construction C f {\displaystyle C_{f}} of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated C f {\displaystyle Cf} . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder M f {\displaystyle Mf} , with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces.
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https://en.wikipedia.org/wiki/Homotopy_cofiber
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In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and k, with k = n − 4. In some older books and papers, E2 and E4 are used as names for G2 and F4.
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https://en.wikipedia.org/wiki/En_(Lie_algebra)
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In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. A quasigroup with an identity element is called a loop.
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https://en.wikipedia.org/wiki/Quasigroup
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In mathematics, especially in algebraic K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen.
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https://en.wikipedia.org/wiki/K-group_of_a_field
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In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another. Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent sheaves and their cohomology.
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https://en.wikipedia.org/wiki/Coherent_sheaf_cohomology
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In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles.
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https://en.wikipedia.org/wiki/Quasi-coherent_sheaf
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Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf.
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https://en.wikipedia.org/wiki/Quasi-coherent_sheaf
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In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction.
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https://en.wikipedia.org/wiki/Adjunction_formula_(algebraic_geometry)
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In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0\ } where f is a polynomial of degree 4, such as f ( x , y , z ) = x 4 + y 4 + x y z + z 2 − 1 {\displaystyle f(x,y,z)=x^{4}+y^{4}+xyz+z^{2}-1} . This is a surface in affine space A3.
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https://en.wikipedia.org/wiki/Quartic_surface
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On the other hand, a projective quartic surface is a surface in projective space P3 of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example f ( x , y , z , w ) = x 4 + y 4 + x y z w + z 2 w 2 − w 4 {\displaystyle f(x,y,z,w)=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}} . If the base field is R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } the surface is said to be real or complex respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over C {\displaystyle \mathbb {C} } , and quartic surfaces over R {\displaystyle \mathbb {R} } . For instance, the Klein quartic is a real surface given as a quartic curve over C {\displaystyle \mathbb {C} } . If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.
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https://en.wikipedia.org/wiki/Quartic_surface
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In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology: K n ( X ) → ⊕ p ≥ 0 H D 2 p − n ( X , Q ( p ) ) . {\displaystyle K_{n}(X)\rightarrow \oplus _{p\geq 0}H_{D}^{2p-n}(X,\mathbf {Q} (p)).} Here, X is a complex smooth projective variety, for example. It is named after Alexander Beilinson.
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https://en.wikipedia.org/wiki/Beilinson_regulator
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The Beilinson regulator features in Beilinson's conjecture on special values of L-functions. The Dirichlet regulator map (used in the proof of Dirichlet's unit theorem) for the ring of integers O F {\displaystyle {\mathcal {O}}_{F}} of a number field F O F × → R r 1 + r 2 , x ↦ ( log | σ ( x ) | ) σ {\displaystyle {\mathcal {O}}_{F}^{\times }\rightarrow \mathbf {R} ^{r_{1}+r_{2}},\ \ x\mapsto (\log |\sigma (x)|)_{\sigma }} is a particular case of the Beilinson regulator. (As usual, σ: F ⊂ C {\displaystyle \sigma :F\subset \mathbf {C} } runs over all complex embeddings of F, where conjugate embeddings are considered equivalent.) Up to a factor 2, the Beilinson regulator is also generalization of the Borel regulator.
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https://en.wikipedia.org/wiki/Beilinson_regulator
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In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by Rydh (2010) and studied further by Bhatt & Scholze (2017), who introduced the name v-topology, where v stands for valuation.
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https://en.wikipedia.org/wiki/V-topology
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In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly speaking, for a variety or scheme X, the idea is to consider étale coverings U → X {\displaystyle U\rightarrow X} and to replace each connected component of U and the higher "intersections", i.e., fiber products, U n := U × X U × X ⋯ × X U {\displaystyle U_{n}:=U\times _{X}U\times _{X}\dots \times _{X}U} (n+1 copies of U, n ≥ 0 {\displaystyle n\geq 0} ) by a single point. This gives a simplicial set which captures some information related to X and the étale topology of it. Slightly more precisely, it is in general necessary to work with étale hypercovers ( U n ) n ≥ 0 {\displaystyle (U_{n})_{n\geq 0}} instead of the above simplicial scheme determined by a usual étale cover. Taking finer and finer hypercoverings (which is technically accomplished by working with the pro-object in simplicial sets determined by taking all hypercoverings), the resulting object is the étale homotopy type of X. Similarly to classical topology, it is able to recover much of the usual data related to the étale topology, in particular the étale fundamental group of the scheme and the étale cohomology of locally constant étale sheaves.
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https://en.wikipedia.org/wiki/Étale_homotopy_type
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In mathematics, especially in algebraic number theory, the Hermite–Minkowski theorem states that for any integer N there are only finitely many number fields, i.e., finite field extensions K of the rational numbers Q, such that the discriminant of K/Q is at most N. The theorem is named after Charles Hermite and Hermann Minkowski. This theorem is a consequence of the estimate for the discriminant | d K | ≥ n n n ! ( π 4 ) n / 2 {\displaystyle {\sqrt {|d_{K}|}}\geq {\frac {n^{n}}{n! }}\left({\frac {\pi }{4}}\right)^{n/2}} where n is the degree of the field extension, together with Stirling's formula for n!. This inequality also shows that the discriminant of any number field strictly bigger than Q is not ±1, which in turn implies that Q has no unramified extensions.
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https://en.wikipedia.org/wiki/Hermite–Minkowski_theorem
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In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y. More generally, in category theory, any functor by definition provides an induced morphism in the target category for each morphism in the source category. For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from (e.g.) the category of topological spaces to (e.g.) the category of groups or rings. This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism.
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https://en.wikipedia.org/wiki/Induced_homomorphism
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A homomorphism induced from a map h {\displaystyle h} is often denoted h ∗ {\displaystyle h_{*}} . Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are inverse to each other up to homotopy induce homomorphisms that are inverse to each other.
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https://en.wikipedia.org/wiki/Induced_homomorphism
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A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations between spaces and continuous maps, often very intricate, can be inferred from relations between the homomorphisms they induce. The latter may be simpler to analyze, since they involve algebraic structures which can be often easily described, compared, and calculated in.
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https://en.wikipedia.org/wiki/Induced_homomorphism
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In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle {\text{Ho}}({\textbf {Top}})} . The main idea is this: if we have a diagram F: I → Top {\displaystyle F:I\to {\textbf {Top}}} considered as an object in the homotopy category of diagrams F ∈ Ho ( Top I ) {\displaystyle F\in {\text{Ho}}({\textbf {Top}}^{I})} , (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone Holim ← I ( F ): ∗ → Top Hocolim → I ( F ): ∗ → Top {\displaystyle {\begin{aligned}{\underset {\leftarrow I}{\text{Holim}}}(F)&:*\to {\textbf {Top}}\\{\underset {\rightarrow I}{\text{Hocolim}}}(F)&:*\to {\textbf {Top}}\end{aligned}}} which are objects in the homotopy category Ho ( Top ∗ ) {\displaystyle {\text{Ho}}({\textbf {Top}}^{*})} , where ∗ {\displaystyle *} is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category Ho ( Top ) {\displaystyle {\text{Ho}}({\textbf {Top}})} since the latter homotopy functor category has functors which picks out an object in Top {\displaystyle {\text{Top}}} and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which give techniques for constructing homotopy limits and colimits in terms of other homotopy categories, such as derived categories. Another perspective formalizing these kinds of constructions are derivatorspg 193 which are a new framework for homotopical algebra.
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https://en.wikipedia.org/wiki/Homotopy_colimit
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In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). In more abstract language, this means that the group homomorphism ρ: G → G L ( V ) {\displaystyle \rho :G\to GL(V)} is injective (or one-to-one).
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https://en.wikipedia.org/wiki/Faithful_representation
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In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function replacing the binary operation; Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group.In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group.
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https://en.wikipedia.org/wiki/Transformation_groupoid
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The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g: A → B {\displaystyle g:A\rightarrow B} , h: B → C {\displaystyle h:B\rightarrow C} , say. Composition is then a total function: ∘: ( B → C ) → ( A → B ) → A → C {\displaystyle \circ :(B\rightarrow C)\rightarrow (A\rightarrow B)\rightarrow A\rightarrow C} , so that h ∘ g: A → C {\displaystyle h\circ g:A\rightarrow C} . Special cases include: Setoids: sets that come with an equivalence relation, G-sets: sets equipped with an action of a group G {\displaystyle G} .Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups.
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https://en.wikipedia.org/wiki/Transformation_groupoid
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In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic example is the category of sets, Set, where the monoidal product of sets A {\displaystyle A} and B {\displaystyle B} is the usual cartesian product A × B {\displaystyle A\times B} , and the internal Hom B A {\displaystyle B^{A}} is the set of functions from A {\displaystyle A} to B {\displaystyle B} . A non-cartesian example is the category of vector spaces, K-Vect, over a field K {\displaystyle K} . Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another.
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https://en.wikipedia.org/wiki/Monoidal_closed_category
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The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations of linguistics; roughly speaking, this is because word-order in natural language matters.
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https://en.wikipedia.org/wiki/Monoidal_closed_category
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In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad to a wide class of functors.
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https://en.wikipedia.org/wiki/Codensity_monad
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In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers.
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https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
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In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes.
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https://en.wikipedia.org/wiki/Minimal_prime_(commutative_algebra)
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In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f: M → N {\displaystyle f:M\to N} between smooth manifolds and S ⊂ M {\displaystyle S\subset M} a closed Whitney stratified subset, if f | S {\displaystyle f|_{S}} is proper and f | A {\displaystyle f|_{A}} is a submersion for each stratum A {\displaystyle A} of S {\displaystyle S} , then f | S {\displaystyle f|_{S}} is a locally trivial fibration. The lemma was originally introduced by René Thom who considered the case when N = R {\displaystyle N=\mathbb {R} } . In that case, the lemma constructs an isotopy from the fiber f − 1 ( a ) {\displaystyle f^{-1}(a)} to f − 1 ( b ) {\displaystyle f^{-1}(b)} ; whence the name "isotopy lemma".
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https://en.wikipedia.org/wiki/Thom's_first_isotopy_lemma
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The local trivializations that the lemma provide preserve the strata. However, they are generally not smooth (not even C 1 {\displaystyle C^{1}} ). On the other hand, it is possible that local trivializations are semialgebraic if the input data is semialgebraic.The lemma is also valid for a more general stratified space such as a stratified space in the sense of Mather but still with the Whitney conditions (or some other conditions).
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https://en.wikipedia.org/wiki/Thom's_first_isotopy_lemma
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The lemma is also valid for the stratification that satisfies Bekka's condition (C), which is weaker than Whitney's condition (B). (The significance of this is that the consequences of the first isotopy lemma cannot imply Whitney’s condition (B).) Thom's second isotopy lemma is a family version of the first isotopy lemma.
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https://en.wikipedia.org/wiki/Thom's_first_isotopy_lemma
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In mathematics, especially in differential topology, Thom's second isotopy lemma is a family version of Thom's first isotopy lemma; i.e., it states a family of maps between Whitney stratified spaces is locally trivial when it is a Thom mapping. Like the first isotopy lemma, the lemma was introduced by René Thom. (Mather 2012, § 11) gives a sketch of the proof. (Verona 1984) gives a simplified proof. Like the first isotopy lemma, the lemma also holds for the stratification with Bekka's condition (C), which is weaker than Whitney's condition (B).
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https://en.wikipedia.org/wiki/Thom_mapping
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In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line ( l ) {\displaystyle (l)} may be studied in isolation —in which case the ambient space of l {\displaystyle l} is l {\displaystyle l} , or it may be studied as an object embedded in 2-dimensional Euclidean space ( R 2 ) {\displaystyle (\mathbb {R} ^{2})} —in which case the ambient space of l {\displaystyle l} is R 2 {\displaystyle \mathbb {R} ^{2}} , or as an object embedded in 2-dimensional hyperbolic space ( H 2 ) {\displaystyle (\mathbb {H} ^{2})} —in which case the ambient space of l {\displaystyle l} is H 2 {\displaystyle \mathbb {H} ^{2}} . To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is R 2 {\displaystyle \mathbb {R} ^{2}} , but false if the ambient space is H 2 {\displaystyle \mathbb {H} ^{2}} , because the geometric properties of R 2 {\displaystyle \mathbb {R} ^{2}} are different from the geometric properties of H 2 {\displaystyle \mathbb {H} ^{2}} . All spaces are subsets of their ambient space.
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https://en.wikipedia.org/wiki/Ambient_space_(mathematics)
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In mathematics, especially in geometry, a double lattice in ℝn is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of any point under the action of a double lattice is a union of two Bravais lattices, related to each other by a point reflection. A double lattice in two dimensions is a p2 wallpaper group. In three dimensions, a double lattice is a space group of the type 1, as denoted by international notation.
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https://en.wikipedia.org/wiki/Double_lattice
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In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.
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https://en.wikipedia.org/wiki/Double_groupoid
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In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their product space X × Y {\displaystyle X\times Y} . In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer. A Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the name has become generic. These many results are named for the German mathematician Hermann Künneth.
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https://en.wikipedia.org/wiki/Kunneth_formula
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In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions Λ i n ⊂ Δ n , 0 ≤ i < n {\displaystyle \Lambda _{i}^{n}\subset \Delta ^{n},0\leq i
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https://en.wikipedia.org/wiki/Fibration_of_simplicial_sets
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In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n×n matrix A = is centrosymmetric when its entries satisfy Ai,j = An−i + 1,n−j + 1 for i, j ∊{1, ..., n}.If J denotes the n×n exchange matrix with 1 on the antidiagonal and 0 elsewhere (that is, Ji,n + 1 − i = 1; Ji,j = 0 if j ≠ n +1− i), then a matrix A is centrosymmetric if and only if AJ = JA.
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https://en.wikipedia.org/wiki/Centrosymmetric_matrix
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In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT): K(m,n) vec(A) = vec(AT) .Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another: vec ( A ) = T {\displaystyle \operatorname {vec} (\mathbf {A} )=^{\mathrm {T} }} where A = . In other words, vec(A) is the vector obtained by vectorizing A in column-major order. Similarly, vec(AT) is the vector obtaining by vectorizing A in row-major order. In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator
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https://en.wikipedia.org/wiki/Commutation_matrix
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In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa.
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https://en.wikipedia.org/wiki/Elimination_matrix
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In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another: Here, a i , j {\displaystyle a_{i,j}} represents the element in the i-th row and j-th column of A, and the superscript T {\displaystyle {}^{\mathrm {T} }} denotes the transpose. Vectorization expresses, through coordinates, the isomorphism R m × n := R m ⊗ R n ≅ R m n {\displaystyle \mathbf {R} ^{m\times n}:=\mathbf {R} ^{m}\otimes \mathbf {R} ^{n}\cong \mathbf {R} ^{mn}} between these (i.e., of matrices and vectors) as vector spaces. For example, for the 2×2 matrix A = {\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}} , the vectorization is vec ( A ) = {\displaystyle \operatorname {vec} (A)={\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}} . The connection between the vectorization of A and the vectorization of its transpose is given by the commutation matrix.
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https://en.wikipedia.org/wiki/Vectorization_(mathematics)
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In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.
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https://en.wikipedia.org/wiki/Galois_connection
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A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections. A Galois connection is rather weak compared to an order isomorphism between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below. The term Galois correspondence is sometimes used to mean a bijective Galois connection; this is simply an order isomorphism (or dual order isomorphism, depending on whether we take monotone or antitone Galois connections).
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https://en.wikipedia.org/wiki/Galois_connection
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In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras. Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales.
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https://en.wikipedia.org/wiki/Complete_Heyting_algebra
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In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S {\displaystyle S} of a preordered set is an element of S {\displaystyle S} which is greater than or equal to any other element of S , {\displaystyle S,} and the minimum of S {\displaystyle S} is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.
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https://en.wikipedia.org/wiki/Maximal_element
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As an example, in the collection ordered by containment, the element {d, o} is minimal as it contains no sets in the collection, the element {g, o, a, d} is maximal as there are no sets in the collection which contain it, the element {d, o, g} is neither, and the element {o, a, f} is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for S .
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https://en.wikipedia.org/wiki/Maximal_element
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{\displaystyle S.} Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banach theorem, the Kirszbraun theorem, Tychonoff's theorem, the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field.
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https://en.wikipedia.org/wiki/Maximal_element
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In mathematics, especially in order theory, a partially ordered set with a unique minimal element 0 has the disjunction property of Wallman when for every pair (a, b) of elements of the poset, either b ≤ a or there exists an element c ≤ b such that c ≠ 0 and c has no nontrivial common predecessor with a. That is, in the latter case, the only x with x ≤ a and x ≤ c is x = 0. A version of this property for lattices was introduced by Wallman (1938), in a paper showing that the homology theory of a topological space could be defined in terms of its distributive lattice of closed sets. He observed that the inclusion order on the closed sets of a T1 space has the disjunction property. The generalization to partial orders was introduced by Wolk (1956).
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https://en.wikipedia.org/wiki/Disjunction_property_of_Wallman
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In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation. The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol ≤ {\displaystyle \,\leq \,} can be used as the notational device for the relation.
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https://en.wikipedia.org/wiki/Preordered_set
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However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol ≤ {\displaystyle \,\leq \,} may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied.
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https://en.wikipedia.org/wiki/Preordered_set
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In words, when a ≤ b , {\displaystyle a\leq b,} one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or → or ≲ {\displaystyle \,\lesssim \,} is used instead of ≤ . {\displaystyle \,\leq .} To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices.
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https://en.wikipedia.org/wiki/Preordered_set
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The converse is not true: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may contain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components.
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https://en.wikipedia.org/wiki/Preordered_set
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In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent. Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net.
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https://en.wikipedia.org/wiki/Cofinality
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In mathematics, especially in order theory, the greatest element of a subset S {\displaystyle S} of a partially ordered set (poset) is an element of S {\displaystyle S} that is greater than every other element of S {\displaystyle S} . The term least element is defined dually, that is, it is an element of S {\displaystyle S} that is smaller than every other element of S . {\displaystyle S.}
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https://en.wikipedia.org/wiki/Bottom_element
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In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X = ( x i j ) {\displaystyle X=(x_{ij})} of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., ∑ i x i j = ∑ j x i j = 1 , {\displaystyle \sum _{i}x_{ij}=\sum _{j}x_{ij}=1,} Thus, a doubly stochastic matrix is both left stochastic and right stochastic.Indeed, any matrix that is both left and right stochastic must be square: if every row sums to 1 then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal.
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https://en.wikipedia.org/wiki/Bistochastic_matrix
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In mathematics, especially in real algebraic geometry, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set.
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https://en.wikipedia.org/wiki/Semialgebraic_space
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In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra g {\displaystyle {\mathfrak {g}}} attached to a reductive group G. It was introduced by Beilinson & Bernstein (1981). Extensions of this theorem include the case of partial flag varieties G/P, where P is a parabolic subgroup in Holland & Polo (1996) and a theorem relating D-modules on the affine Grassmannian to representations of the Kac–Moody algebra g ^ {\displaystyle {\widehat {\mathfrak {g}}}} in Frenkel & Gaitsgory (2009).
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https://en.wikipedia.org/wiki/Beilinson–Bernstein_localization
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In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f: X → Y {\displaystyle f\colon X\to Y} such that both f and its inverse are monotonic (preserving orders of elements). In the special case when X is totally ordered, monotonicity of f already implies monotonicity of its inverse. One and the same set may be equipped with different orders. Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.
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https://en.wikipedia.org/wiki/Order_type
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In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses. Given a continuous mapping f: X → Y of topological spaces, and the category Sh(–) of sheaves of abelian groups on a topological space. The functors in question are direct image f∗: Sh(X) → Sh(Y) inverse image f∗: Sh(Y) → Sh(X) direct image with compact support f! : Sh(X) → Sh(Y) exceptional inverse image Rf!
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https://en.wikipedia.org/wiki/Image_functors_for_sheaves
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: D(Sh(Y)) → D(Sh(X)).The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "f shriek" or "f lower shriek" and "f upper shriek"—see also shriek map. The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to étale sheaves on schemes.
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https://en.wikipedia.org/wiki/Image_functors_for_sheaves
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In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point.
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https://en.wikipedia.org/wiki/Splitting_lemma_(functions)
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In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named after Hans Hahn.
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https://en.wikipedia.org/wiki/Hahn_embedding_theorem
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In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups, (Fine, Rosenberger & Stille 1995). They were introduced in (Nielsen 1921) to prove that every subgroup of a free group is free (the Nielsen–Schreier theorem), but are now used in a variety of mathematics, including computational group theory, k-theory, and knot theory. The textbook (Magnus, Karrass & Solitar 2004) devotes all of chapter 3 to Nielsen transformations.
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https://en.wikipedia.org/wiki/Nielsen_transformation
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In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A. Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group G. The related but different uniform word problem for a class K of recursively presented groups is the algorithmic problem of deciding, given as input a presentation P for a group G in the class K and two words in the generators of G, whether the words represent the same element of G. Some authors require the class K to be definable by a recursively enumerable set of presentations.
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https://en.wikipedia.org/wiki/Kuznetsov's_theorem
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In mathematics, especially in the area of abstract algebra known as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simple modules, projective modules, and indecomposable modules.
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https://en.wikipedia.org/wiki/Principal_indecomposable_module
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In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring R, the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring. To be left (semi-)hereditary, all (finitely generated) submodules of projective left R-modules must be projective, and similarly to be right (semi-)hereditary all (finitely generated) submodules of projective right R-modules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary and vice versa.
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https://en.wikipedia.org/wiki/Hereditary_ring
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In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings.
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https://en.wikipedia.org/wiki/Semisimple_module
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An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see Semisimple representation.
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https://en.wikipedia.org/wiki/Semisimple_module
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In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in (Baer 1940) and are discussed in some detail in the textbook (Lam 1999, §3). Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules.
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https://en.wikipedia.org/wiki/Baer's_criterion
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Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions.
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https://en.wikipedia.org/wiki/Baer's_criterion
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Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring, may not be injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.
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https://en.wikipedia.org/wiki/Baer's_criterion
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In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra.
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https://en.wikipedia.org/wiki/Noncommutative_polynomial_ring
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In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H ≤ G {\displaystyle H\leq G} such that H has property P. Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.
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https://en.wikipedia.org/wiki/Virtually_nilpotent
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This terminology is also used when P is just another group. That is, if G and H are groups then G is virtually H if G has a subgroup K of finite index in G such that K is isomorphic to H. In particular, a group is virtually trivial if and only if it is finite. Two groups are virtually equal if and only if they are commensurable.
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https://en.wikipedia.org/wiki/Virtually_nilpotent
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In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective.
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https://en.wikipedia.org/wiki/Character_module
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In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that G = H K = { h k: h ∈ H , k ∈ K } and H ∩ K = { e } . {\displaystyle G=HK=\{hk:h\in H,k\in K\}{\text{ and }}H\cap K=\{e\}.} Equivalently, every element of G has a unique expression as a product hk where h ∈ H and k ∈ K. This relation is symmetrical: if K is a complement of H, then H is a complement of K. Neither H nor K need be a normal subgroup of G.
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https://en.wikipedia.org/wiki/Complement_(group_theory)
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In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroup F*, which is generated by the Fitting subgroup and the components of G. For an arbitrary (not necessarily finite) group G, the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of G. For infinite groups, the Fitting subgroup is not always nilpotent. The remainder of this article deals exclusively with finite groups.
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https://en.wikipedia.org/wiki/Generalized_Fitting_subgroup
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In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group G {\displaystyle G} , the holomorph of G {\displaystyle G} denoted Hol ( G ) {\displaystyle \operatorname {Hol} (G)} can be described as a semidirect product or as a permutation group.
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https://en.wikipedia.org/wiki/Holomorph_(mathematics)
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In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. in the study of ordered groups, a Z-group or Z {\displaystyle \mathbb {Z} } -group is a discretely ordered abelian group whose quotient over its minimal convex subgroup is divisible.
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https://en.wikipedia.org/wiki/Z-group
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Such groups are elementarily equivalent to the integers ( Z , + , < ) {\displaystyle (\mathbb {Z} ,+,<)} . Z-groups are an alternative presentation of Presburger arithmetic. occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group.
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https://en.wikipedia.org/wiki/Z-group
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In mathematics, especially in the area of algebra known as module theory, Schanuel's lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of dimension shifting.
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https://en.wikipedia.org/wiki/Schanuel's_lemma
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In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representations of the group. Elements of the representation ring are sometimes called virtual representations. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed fields of characteristic p where the Sylow p-subgroups are cyclic is also theoretically approachable.
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https://en.wikipedia.org/wiki/Representation_ring
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In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.
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https://en.wikipedia.org/wiki/Ore_polynomial
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In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for a ∈ R and s ∈ S, the intersection aS ∩ sR ≠ ∅. A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.
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https://en.wikipedia.org/wiki/Ore_ring
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In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas.
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https://en.wikipedia.org/wiki/Pure_subgroup
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In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group. The concept was generalized to essential submodules.
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https://en.wikipedia.org/wiki/Essential_subgroup
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In mathematics, especially in the area of algebraic topology known as stable homotopy theory, the Adams filtration and the Adams–Novikov filtration allow a stable homotopy group to be understood as built from layers, the nth layer containing just those maps which require at most n auxiliary spaces in order to be a composition of homologically trivial maps. These filtrations, named after Frank Adams and Sergei Novikov, are of particular interest because the Adams (–Novikov) spectral sequence converges to them.
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https://en.wikipedia.org/wiki/Adams_filtration
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In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus which is represented by the following differential equations with respect to the standard angular coordinates ( θ 1 , θ 2 , … , θ n ): {\displaystyle \left(\theta _{1},\theta _{2},\ldots ,\theta _{n}\right):} The solution of these equations can explicitly be expressed as If we represent the torus as T n = R n / Z n {\displaystyle \mathbb {T^{n}} =\mathbb {R} ^{n}/\mathbb {Z} ^{n}} we see that a starting point is moved by the flow in the direction ω = ( ω 1 , ω 2 , … , ω n ) {\displaystyle \omega =\left(\omega _{1},\omega _{2},\ldots ,\omega _{n}\right)} at constant speed and when it reaches the border of the unitary n {\displaystyle n} -cube it jumps to the opposite face of the cube. For a linear flow on the torus either all orbits are periodic or all orbits are dense on a subset of the n {\displaystyle n} -torus which is a k {\displaystyle k} -torus. When the components of ω {\displaystyle \omega } are rationally independent all the orbits are dense on the whole space. This can be easily seen in the two dimensional case: if the two components of ω {\displaystyle \omega } are rationally independent then the Poincaré section of the flow on an edge of the unit square is an irrational rotation on a circle and therefore its orbits are dense on the circle, as a consequence the orbits of the flow must be dense on the torus.
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https://en.wikipedia.org/wiki/Irrational_cable_on_a_torus
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In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot. There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.
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https://en.wikipedia.org/wiki/Invertible_knot
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In mathematics, especially in the areas of abstract algebra dealing with group cohomology or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring of a group and of a subgroup. It thus relates the group cohomology with respect to a group to the cohomology with respect to a subgroup. Shapiro's lemma is named after Arnold S. Shapiro, who proved it in 1961; however, Beno Eckmann had discovered it earlier, in 1953.
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https://en.wikipedia.org/wiki/Shapiro's_lemma
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In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Birkhoff in 1944 and has proved to be a powerful generalization of the notion of direct product.
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https://en.wikipedia.org/wiki/Subdirect_product
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In mathematics, especially in the areas of numerical analysis called numerical partial differential equations, a compact stencil is a type of stencil that uses only nine nodes for its discretization method in two dimensions. It uses only the center node and the adjacent nodes. For any structured grid utilizing a compact stencil in 1, 2, or 3 dimensions the maximum number of nodes is 3, 9, or 27 respectively. Compact stencils may be compared to non-compact stencils. Compact stencils are currently implemented in many partial differential equation solvers, including several in the topics of CFD, FEA, and other mathematical solvers relating to PDE's.
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https://en.wikipedia.org/wiki/Compact_stencil
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In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.
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https://en.wikipedia.org/wiki/Polynomial_expression
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Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety.
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https://en.wikipedia.org/wiki/Polynomial_expression
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In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.
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https://en.wikipedia.org/wiki/Injective_object
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In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions: A possesses no non-trivial (that is, not equal to 1 or 0) idempotent elements; the spectrum of A with the Zariski topology is a connected space.
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https://en.wikipedia.org/wiki/Connected_ring
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In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups (Wehrfritz 1999). Carter (1961) proved that any finite solvable group has a Carter subgroup, and all its Carter subgroups are conjugate subgroups (and therefore isomorphic). If a group is not solvable it need not have any Carter subgroups: for example, the alternating group A5 of order 60 has no Carter subgroups.
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https://en.wikipedia.org/wiki/Carter_subgroup
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Vdovin (2006, 2007) showed that even if a finite group is not solvable then any two Carter subgroups are conjugate. A Carter subgroup is a maximal nilpotent subgroup, because of the normalizer condition for nilpotent groups, but not all maximal nilpotent subgroups are Carter subgroups (Ballester-Bolinches & Ezquerro 2006, p. 100).
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https://en.wikipedia.org/wiki/Carter_subgroup
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For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups. Every subgroup containing a Carter subgroup of a soluble group is also self-normalizing, and a soluble group is generated by any Carter subgroup and its nilpotent residual (Schenkman 1975, VII.4.a). Gaschütz (1962) viewed the Carter subgroups as analogues of Sylow subgroups and Hall subgroups, and unified their treatment with the theory of formations.
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https://en.wikipedia.org/wiki/Carter_subgroup
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In the language of formations, a Sylow p-subgroup is a covering group for the formation of p-groups, a Hall π-subgroup is a covering group for the formation of π-groups, and a Carter subgroup is a covering group for the formation of nilpotent groups (Ballester-Bolinches & Ezquerro 2006, p. 100).
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https://en.wikipedia.org/wiki/Carter_subgroup
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