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

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

References

1. Presentations of Semigroups and Inverse Semigroups Archived 2006-10-11 at the Wayback Machine 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)

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