Clifford semigroup

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

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

References

1. Presentations of Semigroups and Inverse Semigroups section 4.3 Some Results on Clifford Semigroups (accessed on 14 December 2014)
2. Algebraic characterizations of inverse semigroups and strongly regular rings theorem 2 (accessed on 14 December 2014)