High Quality Content by WIKIPEDIA articles! In the theory of semigroups, a weak inverse of an element x in a semigroup (S, ) is an element y such that y x y = y. AN element x of S for which there is an element y of S such that x y x = x is called regular. A Regular semigroup 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 x y x = x and y x y = y then S is called an inverse semigroup. 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 A B and B A 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.