Clifford semigroup

A Clifford semigroup (sometimes also called "inverse Clifford semigroup") is a completely regular inverse semigroup. It is an inverse semigroup with $$xx^{-1}=x^{-1}x$$. Examples of Clifford semigroups are groups and commutative inverse semigroups.

In a Clifford semigroup, $$ xy=yx \leftrightarrow x^{-1}y=yx^{-1}$$.