Johnson scheme

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

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