Semigroup with involution: Difference between revisions

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The first of these looks like the definition of a regular element, but is actually in terms of the involution. Likewise, the second axiom appears to be describing the commutation of two idempotents. It is known however that regular semigroups do not form a variety because their class does not contain [[free object]]s (a result established by [[D. B. McAlister]] in 1968). This line of reasoning motivated Nordahl and Scheiblich to begin in 1977 the study of the (variety of) *-semigroups that satisfy only the first these two axioms; because of the similarity in form with the property defining regular semigroups, they named this variety regular *-semigroups.

It is a simple calculation to establish that a regular *-semigroup is also a regular semigroup because ''x''* turns out to be an inverse of ''x''. The rectangular band from [[#ex7|exampleExample 7]] is a regular *-semigroup that is not an inverse semigroup.<ref name="Nordahl and Scheiblich"/> It is also easy to verify that in a regular *-semigroup the product of any two projections is an idempotent.<ref>Nordahl and Scheiblich, Theorem 2.5</ref> In the aforementioned rectangular band example, the projections are elements of the form (''x'', ''x'') and [like all elements of a band] are idempotent. However, two different projections in this band need not commute, nor is their product necessarily a projection since (''a'', ''a'')(''b'', ''b'') = (''a'', ''b'').

Semigroups that satisfy only ''x''** = ''x'' = ''xx''*''x'' (but not necessarily the antidistributivity of * over multiplication) have also been studied under the name of [[I-semigroup]]s.