Injectives in Finitely Generated Universal Horn Classes

Let K be a finite set of finite structures. We give a syntactic characterization of the property: every element of K is invective in ISP(K). We use this result to establish that 4 is invective in ISP(s) for every two-element algebra i. ?0. Introduction. Let K be a finite set of finite structures for a first-order language. In this paper we give a syntactic characterization (Theorem 4) of the property that each member of K is infective in ISP(K), the universal Horn class generated by K. We then show that K = {J} has this property for every two-element algebra J. This paper was motivated by the following question: for which two-element algebras J does ISP(.4) have the amalgamation property? The property stated above is stronger than the amalgamation property, so the answer is: for every twoelement algebra. Model companions appear in Lemma 1. For their definition and elementary properties see [8], but note that we replace a theory by its class of models. Theorem 9 rests on E. Post's classification [9] of all two-valued clones of operations, a summary of which appears in [7]. The results of ?1 are joint work of the authors, the individual contributions being inextricably combined but of equal magnitude. ?2 is due to R. Willard, and the details of ?3 were also worked out by R. Willard based on an explanation of the results in [1] given by M. H. Albert. ?1. The syntactic characterization. Let M be a class of structures for some firstorder language. The classes I(M), S(M), P(M) and Pfin(M) are the closures of M under isomorphism, substructures, products, and finite products respectively. Mfin is the class of all finite members of M. Arrows and hooked arrows between members of M denote homomorphisms and embeddings respectively. A member 9 of M is invective in M if every diagram 1 (a) in M can be completed to a commuting diagram 1(b). Received April 28, 1986; revised October 17, 1986. ' Research supported by an NSERC postdoctoral fellowship. 2Research supported by an Ontario graduate scholarship. ? 1987, Association for Symbolic Logic 0022-4812/87/5203-001 5/$0 1.70