Theory of semigroups

In the theory of semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

s, a weak inverse of an element x in a semigroup is an element y such that .

An element x of S for which there is an element y of S such that is called regular. A regular semigroup

Regular semigroup

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 structure is particularly amenable to study via Green's relations.- Origins...

 is a semigroup in which every element is regular.

If every element x in S has a unique inverse y in S in the sense that and then S is called an inverse semigroup

Inverse semigroup

In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...

.

Category theory

In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a weak inverse of an object A in a monoidal category

Monoidal category

In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...

 C with monoidal product ⊗ and unit object I is an object B such that both A⊗B and B⊗A are isomorphic

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 to the unit object I of C. A monoidal category in which every morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

 is invertible and every object has a weak inverse is called a 2-group

2-group

In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of n-groups...

.

See also

• Generalized inverse
  Generalized inverse
  In mathematics, a generalized inverse or pseudoinverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them...
  
  
• Von Neumann regular ring
  Von Neumann regular ring
  In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...