Unions of relational systems
H. Jerome Keisler · Proceedings of the American Mathematical Society · 1964
We shall assume throughout that K 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 e3 is said to be a union of a set M of structures if each W{ E M is a substructure of e and every element of e3 is an element of some WEM. Our purpose in this paper is to prove the result below. The following twzeo conditions are equivalent: (a,) if e is a union of some subset M of K, then CEK; (b1) K is the class of all models of some set of sentences of the form