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Varieties generated by free completely simple semigroups. (English) Zbl 0663.20064

The class \({\mathcal C}{\mathcal S}\) of all completely simple semigroups is a variety of unary semigroups defined by the identities; \(x=xx^{-1}x\), \(x=(x^{-1})^{-1}\), \(xx^{-1}=x^{-1}x\), and \(xx^{- 1}=(xyx)(xyx)^{-1}\). The author shows that the subvariety \({\mathcal V}_ n\) of \({\mathcal C}{\mathcal S}\) generated by the free completely simple semigroup \(F_ n\) of rank n satisfies a non-trivial identity in \(n+1\) variables and that the chain \({\mathcal V}_ 1\subseteq {\mathcal V}_ 2\subseteq...\subseteq {\mathcal V}_ n\subseteq..\). forms a strictly increasing sequence. A basis of identities for \({\mathcal V}_ 2\) is found (\({\mathcal V}_ 1\) being the variety of abelian groups) from which it is proved that \({\mathcal V}_ 2\) lies properly between \({\mathcal C}\), the variety of central completely simple semigroups (members of \({\mathcal C}{\mathcal S}\) for which the product of two idempotents always lies in the centre of its \({\mathcal H}\)-class) and \({\mathcal D}\), the variety of \({\mathcal C}{\mathcal S}\) semigroups whose idempotent generated subsemigroups have abelian subgroups. Rees matrix representations of the free objects on a countably infinite set are given for the variety \({\mathcal D}\) and for the variety generated by \({\mathcal V}_ 2\).
Reviewer: P.M.Higgins

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
08B15 Lattices of varieties
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References:

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