Semigroup with involution: Difference between revisions

Revision as of 17:19, 7 January 2025

Revision as of 14:46, 24 March 2025

Line 49:

=== 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]] showed that the following two axioms provide an analogous characterization of inverse semigroups as a [[Variety (universal algebra)|subvariety]] of *-semigroups:{{cn|date=Match 2025}}

* {{math|''x'' {{=}} ''xx''*''x''}}

* {{math|(''xx''*)(''x''*''x'') {{=}} (''x''*''x'')(''xx''*)}}

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).{{cn|date=March 2025}} 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|Example 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 {{math|(''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 {{math|(''a'', ''a'')(''b'', ''b'') {{=}} (''a'', ''b'')}}.

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

====P-systems====

The problem of characterizing when a regular semigroup is a regular *-semigroup (in the sense of Nordahl & Scheiblich) wascan be addressed by M. Yamada (1982). He defineddefining a '''P-system'''. F(S) as subset ofFor the idempotents ofsemigroup {{mvar|S}}, denoted as usual bylet {{math|''E''(''S'').}} Usingdenote the usualset notationof idempotents, and let {{math|''V''(''a'')}} fordenote the inverses of ''{{mvar|a}}. A P-system {{math|'', F''(''S'')}} needsis then a subset of {{math|''E''(''S'')}} towhich satisfysatisfies the following axioms:

# For any ''{{mvar|a''}} in {{mvar|S}}, there exists a unique {{math|''a''°}} in {{math|''V''(''a'')}} such that {{math|''aa''°}} and {{math|''a''°''a''}} are in {{math|''F''(''S'')}}

# For any ''{{mvar|a''}} in {{mvar|S}}, and {{mvar|b}} in {{math|''F''(''S'')}}, ''a°ba'' is in F(S), where ° is the well-defined operation from the previous axiom

# For any ''{{mvar|a''}}, ''{{mvar|b''}} in {{math|''F''(''S'')}}, {{math|''ab''}} is in {{math|''E''(''S'')}}; note: not necessarily in {{math|''F''(''S'')}}

A regular semigroup {{mvar|S}} is a *-regular semigroup, as defined by Nordahl & Scheiblich, if and only if it has a p-system {{math|''F''(''S'')}}.<ref>{{harvcoltxt|Yamada|1982}}</ref> In this case {{math|''F''(''S'')}} is the set of projections of {{mvar|S}} with respect to the operation {{math|°}} defined by {{math|''F''(''S'')}}. In an [[inverse semigroup]] the entire semilattice of idempotents is a pP-system. Also, if a regular semigroup {{mvar|S}} has a pP-system that is multiplicatively closed (i.e. subsemigroup), then {{mvar|S}} is an inverse semigroup. Thus, a pP-system may be regarded as a generalization of the semilattice of idempotents of an inverse semigroup.

===*-regular semigroups (Drazin)===