Model companion properties of some theories

Abstract

The class K of algebraic systems of signature is called a formula-definable class if there exists an algebraic system A of signature such that for any algebraic system B of signature it is B 2 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.