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On permutability in semigroup varieties. (English) Zbl 0746.20034

Varieties of congruence permutable semigroups were described in a previous paper by the author [Czech. Math. J. 40(115), No. 3, 441-452 (1990; Zbl 0731.20039)]. The present paper characterizes all varieties of semigroups (varieties of regular *-semigroups) in which each semigroup (regular *-semigroup) with one or two generators (with two generators, resp.) is congruence permutable. The symbol \(W(p=q)\) \((W^*(p=q))\) denotes the variety of all semigroups (variety of all regular *- semigroups, resp.) satisfying the identity \(p=q\).
Results: (1) Let \(\mathcal V\) be a variety of semigroups. Then (a) each \(S\in {\mathcal V}\) with one generator is congruence permutable iff \({\mathcal V}\subseteq W(x=x.x^ n)\) or \({\mathcal V}\subseteq W(x^ n=x.x^ n)\) for a positive integer \(n\); (b) each \(S\in{\mathcal V}\) with two generators is congruence permutable iff \({\mathcal V}\subseteq W(x=x.x^ n)\cap W((x.y.x)^ n=x^ n)\) for a positive integer \(n\). (2) Let \(\mathcal V\) be a variety of regular *-semigroups. Then \(\mathcal V\) is congruence permutable iff each \(S\in {\mathcal V}\) with two generators is congruence permutable iff \({\mathcal V}\subseteq W^*(x.x^*=y.y^*)\).
Reviewer: J.Duda (Brno)

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B10 Congruence modularity, congruence distributivity

Citations:

Zbl 0731.20039
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