On the equational theories of semigroup varieties (Q2755743)

\textit{V. L. Murskiĭ} [Mat. Zametki 3, 663--670 (1968; Zbl 0181.01401)] invented a powerful method to encode defining relations of a semigroup presentation in semigroup identities and, departing from a suitable finitely presented semigroup with undecidable word problem, constructed a finitely based semigroup variety with undecidable equational theory. In the paper under review, Murskiĭ's encoding is applied to a suitable recursive family of finitely presented semigroups in which the semigroups with undecidable word problem form a recursively enumerable non-recursive subfamily. This leads to a recursive family of finitely based semigroup varieties in which the varieties with undecidable equational theory form a recursively enumerable non-recursive subfamily. Thus, the property of a finite set of semigroup identities to define a variety with decidable equational theory is shown to be undecidable.