In mathematics, the term weak inverse is used with several meanings.
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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 y•x•y = 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 x ∈ S, there exists y ∈ S such that xy and yx are idempotents.
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. This is a stronger notion than weak inverse. Every E-inversive semigroup is regular, but not vice versa.
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.
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 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.