ALGEBRAIC CHARACTERISTICS OF THE CRITERION OF COMPLETENESS OF A CLASS OF ALGEBRAIC SYSTEMS

Abstract

In many sources on model theory, in addition to the proven properties about classes of algebraic systems, the characteristics of these properties are given in algebraic terms, that is, they show the nature of these properties from the perspective of universal algebra. For example, the class of quasi-varieties or varieties is defined using model-theoretic concepts, fulfillment of quasi-identities or identities, and in algebraic concepts of closedness with respect to direct products, ultraproducts, fulfillment of locality, closedness with respect to homomorphisms. H.J. Keisler gave an algebraic characterization of the criterion for the axiomatizability of a class of algebraic systems, using the closure of the class under the ultraproduct and isomorphism of algebraic systems, as well as the closure under ultrapowers to complement the class. H.J. Keisler, however, does not give any algebraic characterization of the criterion for completeness of a class of algebraic systems.In this article, an algebraic characterization of the completeness criterion for a class of algebraic systems is obtained. For comparison, it is not possible to give an algebraic description of the criterion for the model completeness of a class, in terms used in the article. This shows that the algebraic nature of complete and model complete classes is somewhat different.