 ## Integral closure of an ideal

In algebra, the integral closure of an ideal I of a commutative ring R, denoted by I ¯ {\displaystyle {\overline {I}}}, is the set of all elements r in R that are integral over I: there exist a i ∈ i {\displaystyle a_{i}\in I^{i}} such that r n + ...

## Inverse semigroup

In group theory, an inverse semigroup is a semigroup in which every element x in S has a unique inverse y in S in the sense that x = xyx and y = yxy, i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear ...

## J-structure

In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by Springer to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion a ...

## Jonsson–Tarski algebra

In mathematics, a Jonsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X × X. They were introduced by Bjarni Jonsson and Alfred Tarski. Smirnov, named them after Georg Can ...

## Kasch ring

In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is defined, and the two properties are independe ...

## Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.

## Matrix semiring

In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication. The set of n × n matrices with entries from R is a matrix ring denoted M n, as well as some subset ...

## Monogenic semigroup

The monogenic semigroup generated by the singleton set { a } is denoted by ⟨ a ⟩ {\displaystyle \langle a\rangle }. The set of elements of ⟨ a ⟩ {\displaystyle \langle a\rangle } is { a, a 2, a 3.}. There are two possibilities for the monogenic s ...

## Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity. They occur in several branches of mathematics. For example, ...

## Monus

In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the − sym ...

## Multiplicative group

In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to ...

## N-ary group

In mathematics, and in particular universal algebra, the concept of an n -ary group is a generalization of the concept of a group to a set G with an n -ary operation instead of a binary operation. By an n -ary operation is meant any map f: G n → ...

## Near-field (mathematics)

In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity, and every non-zer ...

## Near-semiring

In mathematics, a near-semiring is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.

## Nowhere commutative semigroup

In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab = ba then a = b. A semigroup is nowhere commutative if and only if any two elements of S are inverses of each other.

## Planar ternary ring

In mathematics, an algebraic structure {\displaystyle } consisting of a non-empty set R {\displaystyle R} and a ternary mapping T: R 3 → R {\displaystyle T\colon R^{3}\to R\,} may be called a ternary system. A planar ternary ring or ternary field ...

## Primitive ring

In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.

## Racks and quandles

In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of knots, they can be viewed as algebraic constructi ...

## Rational monoid

In mathematics, a rational monoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed by a finite transducer: multiplication in such a monoid is "easy", in the sense that it can b ...

## Regular semigroup

In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a, there exists an element x such that axa = a. Regular semigroups are one of the most-studied classes of semigroups, and their structu ...

## Semifield

In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.

## Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplicatively: x y, or simply xy, denotes the result of applying ...

## Semigroup with three elements

In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1, together with the operation of multiplication ...

## Semigroup with two elements

In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five distinct nonisomorphic semigroups having two elements: Z 2, + 2 where Z 2 = {0.1} and "+ 2 is "addition mo ...

## Semiprimitive ring

In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information abou ...

## Semiring

In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally - this originated as a joke, suggesting that rigs are ...

## Special classes of semigroups

In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists ...

## Symmetric inverse semigroup

In abstract algebra, the set of all partial bijections on a set X forms an inverse semigroup, called the symmetric inverse semigroup on X. The conventional notation for the symmetric inverse semigroup on a set X is I X {\displaystyle {\mathcal {I ...

## Torsion-free abelian group

In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only elemen ...

## Trivial semigroup

In mathematics, a trivial semigroup is a semigroup for which the cardinality of the underlying set is one. The number of distinct nonisomorphic semigroups with one element is one. If S = { a } is a semigroup with one element, then the Cayley tabl ...

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows. A category has two basic properties: t ...

## Category (mathematics)

In mathematics, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is ...

## Accessible category

The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" of the operations needed to generate their objects. The theory originates in the work of Grot ...

## Anamorphism

In computer programming, an anamorphism is a function that generates a sequence by repeated application of the function to its previous result. You begin with some value A and apply a function f to it to get B. Then you apply f to B to get C, and ...

## Applied category theory

Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer science, physics, control theory, natural language processing, probability theory and cau ...

## Auto magma object

In mathematics, a magma object, can be defined in any category C {\displaystyle \mathbf {C} } equipped with a distinguished bifunctor ⊗: C × C → C {\displaystyle \otimes:\mathbf {C} \times \mathbf {C} \rightarrow \mathbf {C} }. Since Mag, the cat ...

In category theory, a branch of mathematics, Becks monadicity theorem gives a criterion that characterises monadic functors, introduced by Jonathan Mock Beck in about 1964. It is often stated in dual form for comonads. It is sometimes called the ...

## Bousfield localization

In category theory, a branch of mathematics, a Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences. Bousfield localization is named after ...

## Browns representability theorem

In mathematics, Browns representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a repre ...

## Bundle (mathematics)

In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects ...

## Burnside category

In category theory and homotopy theory the Burnside category of a finite group G is a category whose objects are finite G -sets and whose morphisms are spans of G -equivariant maps. It is a categorification of the Burnside ring of G.

## Cartesian monoidal category

In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal product is the categorical product is called a cartesian monoidal category. Any category with finite products can be thought of as a cartes ...

## Catamorphism

In category theory, the concept of catamorphism denotes the unique homomorphism from an initial algebra into some other algebra. In functional programming, catamorphisms provide generalizations of folds of lists to arbitrary algebraic data types, ...

## Categorical trace

The trace is defined in the context of a symmetric monoidal category C, i.e., a category equipped with a suitable notion of a product ⊗ {\displaystyle \otimes }. An object X in such a category C is called dualizable if there is another object X ∨ ...

## Category algebra

In category theory, a field of mathematics, a category algebra is an associative algebra, defined for any locally finite category and commutative ring with unity. Category algebras generalize the notions of group algebras and incidence algebras, ...

## Center (category theory)

In category theory, a branch of mathematics, the center is a variant of the notion of the center of a monoid, group, or ring to a category.

The codensity monad of a functor G: D → C {\displaystyle G:D\to C} is defined to be the right Kan extension of G along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor T G: C → C. {\displaystyle T ...

## Coherence condition

In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. ...

## Compact object (mathematics)

In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.

## Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as ...