Finiteness properties of varieties and the restriction to finite algebras.
Answering a question asked by the reviewer [Sci. Math. Jpn. 53, No. 1, 171-199 (2001; Zbl 0990.20039)], the author exhibits an example of a semigroup variety \(\mathbf V\) generated by a 7-element monoid such that every finite semigroup in \(\mathbf V\) has a finite identity basis while some subvariety of \(\mathbf V\) is not finitely based (Theorem~4.3). He also constructs two finite monoids \(M_1\) and \(M_2\) with the following properties: each \(M_i\) generates a limit monoid variety while in the semigroup variety generated by each \(M_i\) every proper finitely generated subvariety is finitely based but some proper subvariety is not (Proposition~5.1). For general algebras, the author observes (Theorem~7.1) that a combination of results by \textit{R.~McKenzie} [Int. J. Algebra Comput. 6, No. 1, 49-104 (1996; Zbl 0844.08011)] and \textit{R.~Willard} [Trans. Am. Math. Soc. 349, No. 7, 2755-2774 (1997; Zbl 0870.03010)] leads to the undecidability of the following decision problems: given a finite algebra, decide whether or not the variety it generates is hereditarily finitely based or is hereditarily finitely generated or has only finitely many subvarieties.
hereditarily finitely generated varieties
scientific article; zbMATH DE number 2173802