# Model companion properties of some theories

## DOI:

https://doi.org/10.31489/2024m2/114-123## Keywords:

model companion, pseudofinite theory, formula-definable class, smoothly approximated structure## Abstract

The class K of algebraic systems of signature *σ *is called a formula-definable class if there exists an algebraic system *A *of signature *σ *such that for any algebraic system *B *of signature *σ *it is *B *∈ K if and only if *Th*(*B*) · *Th*(*A*) = *Th*(*A*). The paper shows that the formula-definable class of algebraic systems is idempotently formula-definable and is an axiomatizable class of algebraic systems. Any variety of algebraic systems is an idempotently formula-definite class. If the class K of all existentially closed algebraic systems of a theory *T *is formula-definable, then a theory of the class K is a model companion of the theory *T*. Also, in the paper the examples of some theories on the properties of formula-definability, pseudofiniteness and smoothly approximability of their model companion were discussed.