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Combinatorial Rees-Sushkevich varieties that are cross, finitely generated, or small. (English) Zbl 1192.20044

The author describes an algorithm that, given a set \(\Sigma\) of \(n\) semigroup identities formed by words of length at most \(k\), decides in time \(O(nk)\) if the variety defined by \(\Sigma\) within the class of all combinatorial Rees-Sushkevich semigroup varieties is generated by a finite semigroup or has only finitely many subvarieties. Observe that in general the decidability of the question whether a given finite set of semigroup identities defines a variety generated by a finite semigroup is a longstanding open problem.
The author also shows that if \(\mathbf V\) and \(\mathbf W\) are two combinatorial Rees-Sushkevich varieties, then the meet \(\mathbf V\wedge\mathbf W\) is generated by a finite semigroup whenever so is each of \(\mathbf V\) and \(\mathbf W\), and the join \(\mathbf V\vee\mathbf W\) has only finitely many subvarieties whenever each of \(\mathbf V\) and \(\mathbf W\) has this property. This fails for meets (respectively, joins) of semigroup varieties in general as shown by M. V. Sapir [Semigroup Forum 42, No. 3, 345-364 (1991; Zbl 0741.20042)].

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties

Citations:

Zbl 0741.20042
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References:

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