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On iterated Mal’cev products with a pseudovariety of groups. (English) Zbl 1244.20052

Given a pseudovariety \(\mathbf H\) of groups, the \(\mathbf H\)-kernel of a finite semigroup \(S\) is the subsemigroup \(K_{\mathbf H}(S)\) consisting of all elements of \(S\) related to 1 under every relational morphism from \(S\) to a group in \(\mathbf H\). If the sequence \(K_{\mathbf H}(S)\supseteq K_{\mathbf H}(K_{\mathbf H}(S))\supseteq\cdots\) reaches the subsemigroup \(\langle E(S)\rangle\) generated by the set of all idempotents in \(S\), the semigroup \(S\) said to be \(\mathbf H\)-solvable. For a semigroup \(S\), its maximal subgroup with \(e\in E(S)\) is denoted by \(S_e\).
The authors prove that \(S\) is \(\mathbf H\)-solvable if and only if for each idempotent \(e\in E(S)\), the group \(\langle E(S)\rangle_e\) is \(\mathbf H\)-subnormal in the group \(S_e\) (Theorem 11). Therefore, if the membership in \(\mathbf H\) is decidable, then the question of whether a finite semigroup is \(\mathbf H\)-solvable is always decidable, without any computability hypotheses on the operator \(S\mapsto K_{\mathbf H}(S)\).
There is also an interesting application to iterated Mal’cev products of a given monoid pseudovariety \(\mathbf V\) with a pseudovariety \(\mathbf H\) of groups (Theorem 32): if \(\mathbf V\) contains all finite semilattices, then the iterated product is always local in the sense of B. Tilson [J. Pure Appl.Algebra 48, 83-198 (1987; Zbl 0627.20031)] while the usual Mal’cev product of \(\mathbf V\) with \(\mathbf H\) may not be local in general.

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
20E10 Quasivarieties and varieties of groups

Citations:

Zbl 0627.20031

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References:

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