Clifford semigroup explained

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} .

Notes and References

1. http://www-circa.mcs.st-and.ac.uk/Theses/cacmsc.pdf Presentations of Semigroups and Inverse Semigroups
2. https://gentzen.wordpress.com/2014/12/06/algebraic-characterizations-of-inverse-semigroups-and-strongly-regular-rings/ Algebraic characterizations of inverse semigroups and strongly regular rings