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== Notions of regularity ==
There are two related, but not identical notions of regularity in *-semigroups. They were introduced nearly simultaneously by Nordahl & Scheiblich (1978) and respectively Drazin (1979).<ref>Crvenkovic and Dolinka</ref>
=== Regular *-semigroups (Nordahl & Scheiblich) ===
As mentioned in the [[#Examples|previous examples]], [[inverse semigroup]]s are a subclass of *-semigroups. It is also textbook knowledge that an inverse semigroup can be characterized as a regular semigroup in which any two idempotents commute. In 1963, [[Boris M. Schein]]
* ''x'' = ''xx''*''x''
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