Horn Sentences in Identity Theory

Horn [2] obtained a sufficient condition for an elementary class to be closed under direct product. Chang and Morel [1] showed that this is not a necessary condition. We will show that, if consideration is restricted to identity theory, that is, a first-order predicate calculus with equality but no other relation symbols or operation symbols, Horn's condition is necessary and sufficient. A model W for identity theory consists of a non-empty domain A, but no relations or operations except equality. If I is an index set, and for each i E I, Wi = is a model for identity theory, then the direct product W of the 2ti is a model for identity theory and has domain A, the Cartesian product of the As. The class of systems defined by a sentence is the class of all models of the sentence. A sentence is preserved under direct product if the class of systems it defines is closed under direct product. If we define the product of an empty set of systems to be the trivial system with domain of one element, we note that the class of sentences preserved under direct product of one or more systems contains as a proper subset the class of sentences preserved under direct product of zero or more systems. A Horn sentence is a sentence in prenex conjunctive normal form with at most one positive disjunct in each conjunct. A restricted Horn sentence is a sentence in prenex conjunctive normal form with precisely one positive disjunct in each conjunct. Horn's theorem states that every Horn sentence is preserved under the direct product of one or more systems. It is an obvious corollary of Horn's result that every restricted Horn sentence is preserved under the direct product of zero or more systems. The only properties expressible by sentences of identity theory must concern the cardinal of the domain. For example, the sentence VxVy. x = y