TY - JOUR
AU - Yeshkeyev, A. R.
AU - Kassymetova, M. T.
AU - Ulbrikht, O. I.
PY - 2018/08/22
Y2 - 2024/10/13
TI - On the Jonsson pairs of Abelian groups in the enriched language
JF - Journal of Mathematics, Mechanics and Computer Science
JA - JMMCS
VL - 95
IS - 3
SE - Mathematics
DO - 10.26577/jmmcs-2017-3-466
UR - https://bm.kaznu.kz/index.php/kaznu/article/view/466
SP - 32-49
AB - <p>This paper is devoted to the study of model-theoretic questions of abelian groups in the framework<br>of the study of Jonsson theories. Indeed, the paper shows that Abelian groups with the additional<br>condition of the distinguished predicate satisfy the conditions of Jonssonness and also the<br>perfectness in the sense of the Jonsson theory. We can see that classical examples from algebra<br>such as fields of fixed characteristic, groups, abelian groups, different classes of rings, Boolean<br>algebras, polygons are examples of algebras whose theories satisfy the conditions of Jonssonness.<br>The conditions of Jonssonness are determined very naturally. This the amalgam property and<br>the joint embedding properties, as well as the inductance of the theory under consideration. The<br>study of the model-theoretic properties of the Jonsson theories in the class of abelian groups is<br>a very urgent problem both in the Model Theory itself and in an universal algebra. The Jonsson<br>theories form a rather wide subclass of the class of all inductive theories. But the Jonsson theories<br>under consideration, in general, are not complete. The classical Model Theory mainly deals with<br>complete theories and in the case of the study of Jonsson theories, there is a deficit of a technical<br>apparatus, which at the present time is developed for studying the theoretical-model properties<br>of complete theories. Therefore, the discovery of analogues of such a technique for the study of<br>Jonsson theories has practical significance in this research topic. In this paper, the signature for<br>one place predicate was extended. The elements realizing this predicate form an existentially closed<br>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<br>for complete theories. In this paper we obtain an analogue of the theorem of W. Szmielew on the<br>elementary classification of Abelian groups, and also an analog of the SchrÂ¨oder-Bernstein property<br>for the Jonsson pairs of the theory of Abelian groups. The results obtained show a close connection<br>between the theoretical-model properties of the Jonsson pair and the model-theoretic properties<br>of the center of the Johnson theory under consideration.</p>
ER -