Finite semigroups whose varieties have uncountably many subvarieties
As was discovered by \textit{A.~N.~Trakhtman} [Mat. Zap. 14, No. 3, 138-143 (1988; Zbl 0691.20047)], a finite semigroup may generate a variety with uncountably many subvarieties. For brevity, let us call such a finite semigroup rich. The paper under review aims at a systematic study of rich semigroups. The author produces several interesting new examples. In particular, he shows that there exist rich finitely based semigroups (Trakhtman's semigroup is not finitely based) and that the direct product of a finite group whose variety has only 3 subvarieties with a finite semigroup generating a variety with countably many subvarieties may be rich. The author also finds a complete classification of rich monoids among monoids of index at least 3, or Rees quotients of the free monoid, or orthodox monoids and proves that every inherently non-finitely based semigroup is rich. The paper contains a number of open problems, the most intriguing being the following decidability question: does there exist an algorithm which when presented with an effective description of a finite semigroup \(S\) (by its Cayley table, say), would determine whether or not \(S\) is rich?
inherently non-finitely based semigroups
scientific article; zbMATH DE number 1471826
scientific article