Editing Inverse element (section)

=== In a semigroup ===
{{main|Regular semigroup}}
The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity; that is, in a [[semigroup]].

In a semigroup ''S'' an element ''x'' is called '''(von Neumann) regular''' if there exists some element ''z'' in ''S'' such that ''xzx'' = ''x''; ''z'' is sometimes called a ''[[Generalized inverse|pseudoinverse]]''. An element ''y'' is called (simply) an '''inverse''' of ''x'' if ''xyx'' = ''x'' and ''y'' = ''yxy''. Every regular element has at least one inverse: if ''x'' = ''xzx'' then it is easy to verify that ''y'' = ''zxz'' is an inverse of ''x'' as defined in this section. Another easy to prove fact: if ''y'' is an inverse of ''x'' then ''e'' = ''xy'' and ''f'' = ''yx'' are [[idempotent element|idempotent]]s, that is ''ee'' = ''e'' and ''ff'' = ''f''. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and ''ex'' = ''xf'' = ''x'', ''ye'' = ''fy'' = ''y'', and ''e'' acts as a left identity on ''x'', while ''f'' acts a right identity, and the left/right roles are reversed for ''y''. This simple observation can be generalized using [[Green's relations]]: every idempotent ''e'' in an arbitrary semigroup is a left identity for ''R<sub>e</sub>'' and right identity for ''L<sub>e</sub>''.<ref>Howie, prop. 2.3.3, p. 51</ref> An intuitive description of this fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.

In a monoid, the notion of inverse as defined in the previous section is strictly narrower than the definition given in this section. Only elements in the Green class  [[Green's relations#The H and D relations|''H''<sub>1</sub>]] have an inverse from the unital magma perspective, whereas for any idempotent ''e'', the elements of ''H''<sub>e</sub> have an inverse as defined in this section. Under this more general definition, inverses need not be unique (or exist) in an arbitrary semigroup or monoid. If all elements are regular, then the semigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly one inverse as defined in this section, then the semigroup is called an [[inverse semigroup]]. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an [[absorbing element]] 0 because 000 = 0, whereas a group may not.

Outside semigroup theory, a unique inverse as defined in this section is sometimes called a '''quasi-inverse'''. This is generally justified because in most applications (for example, all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity (see [[Generalized inverse]]).