Mathematical logic, algebra and number theory

On the number of countable models of constant and unary predicates expansions of the dense meet-tree theory

A. B. Dauletiyarova^a, V. V. Verbovskiy<sup>b

a</sup> SDU University Abylai Khan street, 1/1 040900, Kaskelen, Kazakhstan
^b Institute of Mathematics and Mathematical Modeling, Shevchenko street, 28, 050010, Almaty, Kazakhstan

Abstract: In the paper, we investigate Ehrenfeucht theories, that is, theories which have finitely many countable models but which are not countably categorical. More precisely, we count all possible numbers of countable models of the theory DMT of dense meet-trees expanded by several sequences of constants including decreasing ones and by unary predicates with finite realizations. Also, we study the realizations of models over a certain set of formulas based on the Rudin-Keisler preorders on models.

Keywords: Constant expansion, Ehrenfeucht theory, the number of countable models, the number of limit models, the number of prime models, small theory, Rudin-Keisler preorder.

UDC: 510.67

MSC: 06A75, 03C10, 03C15

Received June 30, 2024, published October 23, 2024

Language: English

DOI: 10.33048/semi.2024.21.051