Properties of semigroups of elementary types of model classes
Kabidenov A. Kassatova A. Bekenov M.I. Mamyraly A.
30 September 2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#2025Issue 3142 - 149 pp.
The study of classes of first-order countable language models and their properties is an important direction in model theory. Of particular interest are axiomatizable classes of models (varieties, quasivarieties, finitely axiomatizable classes, Jonssonian classes, etc.). In this paper we present the results obtained on the properties of formula-definable classes of models and formula-definable semigroups of elementary types, namely, we study the properties of semigroups of elementary types of models in a first-order language. We consider products of elementary types which form a commutative semigroup with unit. A two-place relation of absorption of one elementary type by another is introduced, which allows us to distinguish formula-definable semigroups of elementary types and corresponding classes of models. On the basis of the axiomatizability property of formula-definite semigroups of elementary types, their connection with ultraproducts and infinite products is established. Examples of idempotently formula-definite and non-idempotently formula-definite semigroups are given, and their peculiarities are discussed. The paper demonstrates both the study of semigroups of elementary types and the study of properties of formula-definite classes of models.
axiomatizable class , countable signature , elementary types of model classes , formula-definable model , formula-definable semigroups , idempotent , model companion , non-formula-definable model classes , properties of semigroups
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L.N. Gumilyov Eurasian National University
Abai Kazakh National Pedagogical University
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