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=== 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]] has published the following two axioms providing an analogous characterization of inverse semigroups as a [[Variety (universal algebra)|subvariety]] of *-semigroups:
* ''x'' = ''xx''*''x''
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