Criterion for the cosemanticness of the Abelian groups in the enriched signature

Abstract

In the present paper we give a criterion of the cosemanticness relative to the Jonsson spectrum of the model in the class of Abelian groups with a distinguished predicate. This paper is devoted to the study of model-theoretic questions of Abelian groups in the frame of the study of Jonsson theories. Indeed, the paper shows that Abelian groups with the additional condition of the distinguished predicate satisfy conditions of Jonssonness and also the perfectness in the sense of Jonsson theory. It is well known that classical examples from algebra such as fields of fixed characteristic, groups, abelian groups, different classes of rings, Boolean algebras, polygons are examples of algebras whose theories satisfy conditions of Jonssonness. The study of the model-theoretic properties of Jonsson theories in the class of abelian groups is a very urgent problem both in the Model Theory itself and in an universal algebra. The Jonsson theories form a rather wide subclass of the class of all inductive theories. But considered Jonsson theories in general are not complete. The classical Model Theory mainly deals with complete theories and in case of the study of Jonsson theories, there is a deficit of a technical apparatus, which at the present time is developed for studying the modeltheoretic properties of complete theories. Therefore, the finding of analogues of such technique for the study of Jonsson theories has practical significance in thegiven research topic. In this paper the signature for one-place predicate was extended. The elements realizing this predicate form an existentially closed submodel of the considering Jonsson theory's some model. In the final analysis, we obtain the main result of this article as a refinement of the well-known W. Szmielew's theorem on the elementary classification of Abelian groups in the frame of the study of Jonsson theories, thereby the generalization of the well-known question of elementary pairs for complete theories was obtained. Also we obtained the Jonsson analogue for the joint embeddability of two models, or in another way the Schröder-Bernstein properties in the frame of the study of the Johnson pairs of Abelian groups' theory.