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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.