Combinatorial geometries in model theory

Postgraduate

Abstract

Model theory and combinatorial pregeometries are closely related through the so called algebraic closure operator on strongly minimal sets. The study of projective and affine pregeometries are especially interesting since they have a close relation to vectorspaces. In this thesis we will see how the relationship occur and how model theory can conclude a very strong classification theorem which divides pregeometries with certain properties into projective, affine and degenerate (trivial) cases.


Introduction

This research has two major topics woven into each other. The two topics are combinatorial pregeometries (aka matroids) and model theory (in mathematical logic). The understanding of model theory is (as will be shown in this research) very closly connected to the study of pregeometries. Many questions in model theory have been answered by the understanding of pregeometries. Conversely, work in model theory has contributed to the understanding of pregeometries (for instance the classification theorem in this essay).

One of the early topics of model theory was to study totally categorical theories, that is, theories with exactly one model up to isomorphism in every infinite cardinality. This class of theories is quite restrictive, but the ideas that where first developed in this context have had great influence on the further development of model theory. A very important result in this area is due to M.Morley, who proved that theories categorical in one uncountable cardinality is categorical in all uncountable cardinalities. Such theories are called uncountably categorical. It was asked if the uncountably categorical theories were finitely axiomatizable. B.Zilber later showed that totaly categorical theories can’t have this property. This work by Zilber also gave a lot of other interesting information (necessary for his result) about the structure of the models of uncountable categorical theories. The information did in turn generate, together with the so called “stability theory”, also called “classification theory”, a subfield inside model theory called “geometric stability theory”.

To be able to understand the models of uncountably categorical theories (also called uncountably categorical structures), it is vital to understand how the so called minimal subsets are built up. These are the smallest possible infinite definable subsets of the structure’s universe. By using these minimal sets we may prove a lot of important properties of the whole structure. One of the properties of a minimal subset A of a model M is that it induces a combinatorical pregeometry, which pretty much rules M, if M is uncountably categorical or totally categorical. Pregeometries obtained from minimal sets were first studied by M.Marh, and were later generalized by Baldwin and Lachlan.

The pregeometies on minimal sets are one of the connections between combinatorical geometry and model theory. In this research we will among other things study how these pregeometries are created from minimal subsets. We will also look at another combinatorical structure, called pseudoplane, and see how it interacts with the pregeometries when defined in a model of an uncountably categorical theory (Trichotomy theorem 7.1). This result will in turn lead us to the Classification theorem 8.1 of infinite homogenous locally finite, pregeometries, which says that any such pregeometry is either degenerated (trivial) or isomorphic to a projective or affine geometry over a finite field. The presentation of the proof of this theorem is based on B.Zilbers work as presented in. The classification theorem was also proved by Cherlin, Harrington and Lachlan (about the same time as Zilber), but using the classification of finite simple groups. It was later proved, using only combinatorical geometry, by Evans.

An interesting question, which may arrise while reading about the classification theorem, is if homogenity is a necessary assumption. Is there any infinite locally finite non-homogenous pregeometry? The question was open until the 1990s when E.Hrushovski [6], deviced a method based on amalgamation of finite structures, which made it possible to create this kind of pregeometry (which can be found in.

In this research, complete proofs will not be given for all results. Instead we concentrate on explaining the role of the pregeometry obtained by the algebraic closure operator when working on minimal sets. Some very tedious reults in the field of locally projective geometries will also be stated without proofs, but with reference to where they can be found. Zilber’s original proof involves some more advanced model theoretic notions like types, Morley-rang, Morley-degree etc. which we will exclude here. Instead we will emphasise how the results fit together and how the geometric and model theoretic results cooperate.