Clifford semigroup

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

In a Clifford semigroup,^[2] ${\displaystyle xy=yx\leftrightarrow x^{-1}y=yx^{-1}}$.