A basic concept of the theory of models is that of elementary equivalence of similar relational systems: two such systems are said to be elementarily equivalent if they satisfy the same first-order statements or, in other words, if they have the same (first-order) complete theory. It is possible to reformulate this notion of elementary equivalence of systems within the framework of algebraic logic by replacing theories by algebraic structures derived from them or more directly from the systems which are models of these theories. To any such theory T (or model of it), is indeed associated a locally finite polyadic algebra with equality, the underlying Boolean algebra of which is simply the well-known Tarski-Lindenbaum algebra of the theory. It is not hard to prove (see Section 6.1) that two systems are elementarily equivalent iff (i.e. if and only if) they have isomorphic polyadic. algebras. The possibility of replacing theories by algebraic structures and of reducing the purely logical concept of elementary equivalence to the algebraic one of isomorphism can be exploited to give a purely algebraic treatment of model-theoretic problems and suggests natural questions concerning these structures. The present paper illustrates that possibility.

In a work widely quoted and applied,3 Craig has shown that if A and C are any formulas of predicate logic such that A├C, then there is a formula B such that (i) A├B and B├C, and (ii) each predicate symbol occurring in B occurs both in A and in C.4 If, in this theorem, we replace the syntactic notion of derivability, ├, by the semantical notion of consequence, ╞, the resulting proposition is of course equally valid, for by the (strong) completeness theorem of predicate logic5 the relations ├ and ╞ coincide in extension.

Recent interest in Łukasiewicz' Lℵ0, raises the question whether this axiomatic system can be simplified [2]. It is known that Łukasiewicz' fourth axiom CCCPQCQPCQP is dependent [4] p. 51. The axiomatic system resulting from deleting the fourth axiom from Lℵ0 will be shown to be “minimal” in the sense that the axioms and rules of inference are mutually independent; consequently, no further simplification of Lℵ0, is possible. By switching basic operators, however, simplified modifications of Lℵ0, can be constructed.

This paper answers negatively the chief remaining open questions concerning the separation properties of first-order prefix classes of structures (discussed by Addison [2], pp. 26–37).

1. Preliminaries. The first and second separation properties and the reduction property are defined for arbitrary classes of subsets of an arbitrary set.

In [1] Dreben showed that the subclass K′ (described below) of the Suranyi reduction class is recursively solvable by showing that the subclass is finitely controllable; that is, by showing that any member S of K′ is satisfiable only if it is finitely satisfiable. Dreben's argument is very complex, but much of the complexity is due to his proving not merely solvability, but the deeper property of finite controllability. In the present note, by exploiting certain features of Dreben's technique, a simpler, direct proof of the solvability of K′ is obtained — that is, a proof in which the question of satisfiability in a finite domain plays no role.