Existentially closed structures

Abstract

One of the major problems of model theory is the spectrum problem, i.e. the development of structure theorems for the spectrum of models of a given theory. I hope that this paper will make a (small) contribution to the solution of this problem.Broadly speaking, in this paper we take a fixed (but arbitrary) theory T and consider a particular class ℰ of models of T. The structures in ℰ (which are known as existentially closed structures) are connected with the model complete extensions of T. These structures have already appeared several times in the literature.In §1 we survey the notation and terminology that we use, as well as the well known facts that we require. We also consider several concepts which are not new here but which may not be well known.In §2 we define, and give the basic results concerning existentially closed structures. For most theories, T, the class ℰ is not elementary and hence is not directly amenable to a model theoretic study. Because of this we consider a smaller class of structures—the uniformly existentially closed structures. These are discussed in §3.In §4 we look at model complete theories via existentially closed structures. This section is essentially a refinement of parts of [6].In §5 we look at model companions of theories via existentially closed structures.Finally, in §6 we make some remarks on the relevance of existentially closed structures to the spectrum problem.