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On iterated semidirect products of finite semilattices. (English) Zbl 0743.20056

In [Semigroup Forum 28, 73-81 (1984; Zbl 0527.20046)], J.-E. Pin proved that the pseudovariety of monoids generated by all semidirected products of two semilattice monoids is defined by the identities \(xuyvxy=xuyvyx\) and \(xux=xux^ 2\), and later posed the question as to whether, for any \(n\), the pseudovariety of monoids generated by all semidirect products of \(n\) semilattice monoids is also finitely based. In the current paper, the author answers this question in the negative, as a consequence of the more general consideration of the pseudovarieties \({\mathcal S}\ell^ n\) of semigroups, rather than monoids, generated by semidirect products of \(n\) semilattices. He finds a fairly simple basis of identities for \({\mathcal S}\ell^ n\) and shows that, for \(n>2\), no finite number of these identities will suffice. An explicit connection is set up between the monoid and semigroup pseudovarieties of these types. Various other interesting properties of these pseudovarieties are investigated.

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems

Citations:

Zbl 0527.20046
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References:

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