Existential Horn sentences

A. Horn [2] has shown that every sentence of a certain class, which we define precisely below, is true in a direct product of algebras whenever it is true in each component algebra. McKinsey [4] had already shown essentially that every universal sentence preserved under direct product is equivalent to one of these Horn sentences. We establish here the analogous result for existential sentences. This answers a question raised by Chang and Morel [1], who showed that the parallel result does not hold for universal-existential sentences. The problem for existential-universal sentences remains open. An algebraic system A will be taken to consist of a nonempty set of elements, upon which are defined various operations, and various relations in addition to equality. Thus A is a model for a first order language L with symbols for the operations and relations of A. (For details, see [3 ].) Only those models will be considered in which the equality symbol is interpreted by a relation having the usual formal properties of equality-that is, by a congruence on the algebra A; but we do not require that the equality symbol be interpreted by strict identity. The direct product of "similar" algebras, that is, models for the same language, is defined in the usual way. Under this definition the direct product of an empty set of algebras is a trivial algebra, in which all relations, including equality, are universal. A transformation of the prenex conjunctive normal form shows that every formula of L is equivalent to a conditional formula, of the form