Unsolvability of problems of equality and divisibility in certain varieties of semigroups (Q1065939)

The first example of a nontrivial variety of semigroups in which the word problem is unsolvable (that is, unsolvable for at least one finitely presented member of the variety) was given by \textit{V. L. Murskij} [Mat. Zametki 3, 663-370 (1968; Zbl 0181.014)]. A word A is called ''blocking'' if for any finite alphabet the set of words in this alphabet which do not contain values of the word A is finite. A description of blocking words was given by \textit{D. R. Bean}, \textit{A. Ehrenfeucht} and \textit{G. F. McNulty} [Pac. J. Math. 85, 261-294 (1979; Zbl 0428.05001)]. The author proves that any variety defined by identities \(A_ i=B_ i\), \(i=1,2,..\). in finitely many variables, such that for each i the words \(A_ i\) and \(B_ i\) are nonblocking, has unsolvable word problem (and unsolvable divisibility problem). The proof ingeniously associates with any given semigroup S a semigroup \(S^*\), in such a variety, whose word or divisibility problem is solvable if and only if that of S is solvable. For varieties defined by identities \(A_ i=0\), \(i=1,2,..\). in finitely many variables, the converse is also true, that is, if at least one \(A_ i\) is blocking then the word and divisibility problems are solvable. This is not true in general.