ON the D-saturation property

Abstract

In this paper we study the notion of a D-saturated model, which occupies an intermediate position between the notions of a homogeneous model and a saturated model. Both the homogeneous models and the saturated models play a very important role in model theory. For example, the saturated models are the universal domains of the corresponding theories in the sense that is used in algebraic geometry (one can also notice that the algebraically closed fields of infinite transcendence degree, which are the universal domains in algebraic geometry, are the saturated models of the theory of algebraically closed fields); this means that the saturated models realize all what is necessary for the effective study of the corresponding theory.The D-saturated models proved to be useful in various situations. Naturally the question arises on the conditions under which a model is D-saturated. In this paper we indicate some conditions of such sort for weakly o-minimal models and models of stable theories. Namely, for such models we prove that homogeneity and a certain approximation of Dsaturation imply D-saturation. In this paper we study the notion of a D -saturated model, which occupies an intermediate position between the notions of a homogeneous model and a saturated model. Both the homogeneous models and the saturated models play a very important role in model theory. For example, the saturated models are the universal domains of the corresponding theories in the sense that is used in algebraic geometry (one can also notice that the algebraically closed fields of infinite transcendence degree, which are the universal domains in algebraic geometry, are the saturated models of the theory of algebraically closed fields); this means that the saturated models realize all what is necessary for the effective study of the corresponding theory.The D -saturated models proved to be useful in various situations. Naturally the question arises on the conditions under which a model is D -saturated. In this paper we indicate some conditions of such sort for weakly o-minimal models and models of stable theories. Namely, for such models we prove that homogeneity and a certain approximation of D saturation imply D -saturation.