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A Clifford semigroup is a completely regular inverse semigroup.It is an inverse semigroup with x x − 1 = x − 1 x {\displaystyle xx^{-1}=x^{-1}x}. Examples of Clifford semigroups are groups and commutative inverse semigroups.
In a Clifford semigroup, x y = y x ↔ x − 1 y = y x − 1 {\displaystyle xy=yx\leftrightarrow x^{-1}y=yx^{-1}}.
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