Strong Splitting in Stable Homogeneous Models

In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in [Hy]. We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see [Hy]. 1. Basic definitions and spectrum of stability The purpose of this paper is to develop theory of independence for elementary submodels of a homogeneous structure. We get a model class of this kind if in addition to it’s first-order theory we require that the models omit some (reasonable) set of types, see [Sh1]. If the set is empty, then we are in the ’classical situation’ from [Sh2]. In other words, we study stability theory without the compactness theorem. So e.g. the theory of ∆-ranks is lost and so we do not get an independence notion from ranks. Our independence notion is based on strong splitting. It satisfies the basic properties of forking in a rather weak form. The main problem is finding free extensions. So the arguments are often based on the definition of the independence notion instead of the ’independence-calculus’. ∗ Partially supported by the Academi of Finland. † Research supported by the United States-Israel Binational Science Foundation. Publ. 629.