# Weak inverse

In mathematics, the term weak inverse is used with several meanings.

## Theory of semigroups

In the theory of semigroups, a weak inverse of an element x in a semigroup (S, •) is an element y such that yxy = y. If every element has a weak inverse, the semigroup is called an E-inversive or E-dense semigroup. An E-inversive semigroup may equivalently be defined by requiring that for every element xS, there exists yS such that xy and yx are idempotents.[1]

An element x of S for which there is an element y of S such that xyx = x is called regular. A regular semigroup is a semigroup in which every element is regular. This is a stronger notion than weak inverse. Every E-inversive semigroup is regular, but not vice versa.[1]

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

## Category theory

In category theory, a weak inverse of an object A in a monoidal category C with monoidal product ⊗ and unit object I is an object B such that both AB and BA are isomorphic to the unit object I of C. A monoidal category in which every morphism is invertible and every object has a weak inverse is called a 2-group.