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 is an element "y" such that . 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 , there exists such that and are idempotents.
An element "x" of "S" for which there is an element "y" of "S" such that is called regular. A regular semigroup is a semigroup in which every element is regular. This is a stronger notion than weak inverse. Every regular semigroup is "E"-inversive, but not vice versa.
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.
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 and 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.