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Regular semigroup
In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'', there exists an element ''x'' such that ''axa'' = ''a''.〔Howie 1995 : 54.〕 Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.〔Howie 2002.〕
== History ==
Regular semigroups were introduced by J. A. Green in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which Green's relations were introduced. The concept of ''regularity'' in a semigroup was adapted from an analogous condition for rings, already considered by J. von Neumann.〔von Neumann 1936.〕 It was Green's study of regular semigroups which led him to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees.
The term inversive semigroup (French: demi-groupe inversif) was historically used as synonym in the papers of Gabriel Thierrin (a student of Paul Dubreil) in the 1950s,〔http://www.csd.uwo.ca/~gab/pubr.html〕 and it is still used occasionally.
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