Abstract

Saturated models are a powerful tool in model theory. The properties of universality and homogeneity of the saturated models of a theory are useful for proving facts about this theory. They are used in the proof of interpolation and preservation theorems and also as work-spaces. Sometimes we work with models which are saturated only for some sets of formulas, for example, recursively saturated models, in the study of models of arithmetic or atomic compact, in model theory of modules. In this article we introduce the notion of equality-free saturated model, that is, roughly speaking, a model which is saturated for the set of equality-free formulas. Our aim is to understand better the role that identity plays in classical model theory, in particular with regard to this process of saturation.