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Axiomatizable classes with strong homomorphisms. (English) Zbl 0639.03035
The author offers as solution of Problem 1 in § 3.1 of 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 \({\mathcal K}.]\)
Reviewer: G.Fuhrken
03C52 Properties of classes of models
08C10 Axiomatic model classes
03C60 Model-theoretic algebra
Full Text: DOI
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