Finiteness properties of varieties and the restriction to finite algebras. (Q1780019)

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.