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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 ''x'' ∈ ''S'', there exists ''y'' ∈ ''S'' such that ''xy'' and ''yx'' are idempotents.〔 (preprint )〕
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 ''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 and then ''S'' is called an inverse semigroup.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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