On the Jonsson pairs of Abelian groups in the enriched language

Abstract

This paper is devoted to the study of model-theoretic questions of abelian groups in the framework
of the study of Jonsson theories. Indeed, the paper shows that Abelian groups with the additional
condition of the distinguished predicate satisfy the conditions of Jonssonness and also the
perfectness in the sense of the Jonsson theory. We can see 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 the conditions of Jonssonness.
The conditions of Jonssonness are determined very naturally. This the amalgam property and
the joint embedding properties, as well as the inductance of the theory under consideration. The
study of the model-theoretic properties of the 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 the Jonsson theories
under consideration, in general, are not complete. The classical Model Theory mainly deals with
complete theories and in the 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 theoretical-model properties
of complete theories. Therefore, the discovery of analogues of such a technique for the study of
Jonsson theories has practical significance in this research topic. In this paper, the signature for
one place predicate was extended. The elements realizing this predicate form an existentially closed
submodel of some model of the Jonsson theory under consideration. As a result, we have the Jonsson generalization of the well-known problem of elementary pairs
for complete theories. In this paper we obtain an analogue of the theorem of W. Szmielew on the
elementary classification of Abelian groups, and also an analog of the Schr¨oder-Bernstein property
for the Jonsson pairs of the theory of Abelian groups. The results obtained show a close connection
between the theoretical-model properties of the Jonsson pair and the model-theoretic properties
of the center of the Johnson theory under consideration.