Unions of Relational Systems

We shall assume throughout that if is a class of structures (i.e., relational systems) that is elementary in the wider sense; that is, K is the class of all models of some finite or infinite set of sentences of the first order predicate logic with identity. A structure 23 is said to be a union of a set M of structures if each 21G M is a substructure of 23 and every element of 23 is an element of some 21E M. Our purpose in this paper is to prove the result below. The following two conditions are equivalent : (ai) if 23 is a union of some subset M of K, then 23 EK; (bi) K is the class of all models of some set of sentences of the form