Properties of semigroups of elementary types of model classes

Abstract

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 formuladefinite 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.