Axiomatizable classes with strong homomorphisms (Q1100195)

The author offers as solution of Problem 1 in {\S} 3.1 of \textit{A. I. Mal'tsev}'s paper in Trud. Chetvert. Vsesoyuzn. Mat. S''ezda, Leningrad, 3-12 Iyulya 1961, 1, 169-198 (1963; Zbl 0191.295), the following theorem: All homomorphisms between members of an axiomatic class \({\mathcal K}\) are strong homomorphisms if and only if for each predicate there is an ``S-axiom'' true in all members of \({\mathcal K}\). For the intended definition of ``S-axiom'' the reader should look at the sufficiency part of the proof on page 118, rather than to the author's definition on page 115. [The reviewer would have understood Mal'tsev's problem to refer to all homomorphisms from members of \({\mathcal K}\) and not just to those whose target is also in \({\mathcal K}.]\)