Regular semigroup

Summary

A regular semigroup is a semigroup in which each element $a$ has a partner $x$ with $axa = a$. Groups, the full transformation semigroups, and the semigroups of all $n \times n$ matrices over a field under multiplication are regular. Regular semigroups are the class in which Green's relations give the most detailed structure theory. The inverse semigroups are the regular semigroups in which each $a$ has exactly one $x$ with $axa = a$ and $xax = x$.