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By AnatoliГ­ IvanoviДЌ Mal'cev (Eds.)

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If is any model in 72(M),and u is a homomorphism of m onto %, then W < mu. J) be similar models. U? onto '21". The maps u" naturally induce a map u from fm into % . Let 7C be an abstract class of models. , members of 7C) at least one of the projections is an isomorphism. ml Theorem 2: An abstract class 7C of models contains all subdirect products of its members i f f for every set M, the set %(M) of %-modelswith base M forms a complete lower subsemilattice of the lattice W(M)of all models with base M of the type of %(in other words, %(M) is closed under arbitrary products - in the sense of %(M) - of its members).

3) Some plausible clarifications: The predicate symbols permissible in Sz are those associated with the basic predicates of % (operation symbols can appear if ')r is an algebra), arbitrary symbols with arguments restricted to the auxiliary set, equality, and E, which alone has mixed arguments. A special (cf. [Rl]), two-sorted form of Skolem's theorem is required. The second sentence in the proof should follow the fifth. Then in the reduc- 26 Representations of models tion process all other subformulas involving individual constants from A can be eliminated in favor of T or F as determined by the diagram of a.

The collection H of all homomorphisms of %-structures is a category in the stated sense. Objects of this category will be identified with the structures of which they in fact are the identity map;; in this way, the class 3c of structures can be viewed as a category. In a category % of structures, a structure % is called a substructure of the ) ; a homomorphism of 9 into an ar%-structure % iff: (1) p ( 8 ) C ~ ( 8 (2) bitrary %'-structure 0.

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