Olivier Lessmann

Shelah's stability spectrum and homogeneity spectrum in finite diagrams

Abstract. We present Saharon Shelahs Stability Spectrum and Homogeneity Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new.

A PRIMER OF SIMPLE THEORIES

Abstract. We present a self-contained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelahs 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.

Ranks and pregeometries in finite diagrams

Abstract The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D . In this work, we introduce a natural dependence relation on the subsets of the models for the ℵ 0 -stable case which share many of the formal properties of forking. This is achieved by considering a rank for this framework which is bounded when the diagram D is ℵ 0 -stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental first-order theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results (categoricity) as well as negative results (construction of nonisomorphic models). We also give a proof of a Two Cardinal Theorem for this context.

Amalgamation properties and finite models in Ln-theories

Abstract. Djordjević [Dj 1] proved that under natural technical assumptions, if a complete Ln-theory is stable and has amalgamation over sets, then it has arbitrarily large finite models. We extend his study and prove the existence of arbitrarily large finite models for classes of models of Ln-theories (maybe omitting types) under weaker amalgamation properties. In particular our analysis covers the case of vector spaces.

Local order property in nonelementary classes

Abstract. We study a local version of the order property in several frameworks, with an emphasis on frameworks where the compactness theorem fails: (1) Inside a fixed model, (2) for classes of models where the compactness theorem fails and (3) for the first order case. Appropriate localizations of the order property, the independence property, and the strict order property are introduced. We are able to generalize some of the results that were known in the case of local stability for the first order theories, and for stability for nonelementary classes (existence of indiscernibles, existence of averages, stability spectrum, equivalence between order and instability). In the first order case, we also prove the local version of Shelahs Trichotomy Theorem. Finally, as an application, we give a new characterization of stable types when the ambient first order theory is simple.