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Download File ASPECTS OF STABILITY IN SIMPLE THEORIES A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Donald A. Brower Steven Buechler, Director Graduate Program in Mathematics Notre Dame, Indiana April 2012 c Copyright by Donald A. Brower 2012 All Rights Reserved ASPECTS OF STABILITY IN SIMPLE THEORIES Abstract by Donald A. Brower Simple theories are a strict extension of stable theories for which non-forking independence is a nice independence relation. However, not much is known about how the simple unstable theories differ from the strictly stable ones. This work looks at three aspects of simple theories and uses them to give a better picture of the differences between the two classes. First, we look at the property of weakly eliminating hyperimaginaries and show that it is equivalent to forking and thorn- forking independence coinciding. Second, we look at the stable forking conjecture, a strong statement asserting that simple unstable theories have an essentially stable “core”, and prove that it holds between elements having SU-rank 2 and finite SU-rank. Third, we consider a property on indiscernible sequences that is known to hold in every stable theory, and show it holds on, at most, a subset of simple theories out of all possible first order theories. CONTENTS ACKNOWLEDGMENTS . iii CHAPTER 0: INTRODUCTION AND BACKGROUND . 1 0.1 Preliminaries . 5 CHAPTER 1: DWIP . 13 1.1 DWIP and the Weak Elimination of Hyperimaginaries . 13 1.2 WEHI and Thorn Forking . 15 1.2.1 Comparison with Ealy’s approach . 15 1.2.2 Non-forking independence has the intersection property . 18 1.3 Equivalence of Forking and Thorn Forking Imply WEHI . 20 CHAPTER 2: SOME STABLE FORKING . 25 2.1 Symmetry of Stable Forking . 26 2.2 Quasidesigns have stable forking sometimes . 29 2.3 Stable Forking with Elements of SU-rank ≤ 2 ........... 33 2.4 A broader application of DWIP . 36 CHAPTER 3: SEQUENCE EXTENSIONS . 38 3.1 Common Extensions . 39 3.2 Graph Relation from No Common Extension . 44 3.3 Small-Tree Property . 47 3.4 Fans .................................. 52 3.5 A Small Tree Example . 55 BIBLIOGRAPHY . 58 ii ACKNOWLEDGMENTS I would like to thank all the people who helped me through my graduate years. Special thanks go to Cameron D. Hill and Sergei Starchenko. I would like to thank my advisor Steve Buechler for his assistance, and the Schmitt Foundation for a generous fellowship. Also, the loving support of Sarah, Chauncey, and my family. iii CHAPTER 0 INTRODUCTION AND BACKGROUND Simple theories are a class of elementary first-order theories which strictly in- clude the class of stable theories. They are characterized by the nice behavior of non-forking independence over arbitrary subsets. While their original uncovering was by Shelah (see [18]) in the course of looking for theories with nice combina- torial properties, interest waned until Kim (in [7]) showed that non-forking inde- pendence extended virtually unchanged from stable theories. Over the following years many properties of stable theories were reproved in the simple context. Two examples are the orthogonality of types and the supersimple/simple dichotomy. Other objects were extended, albeit with some changes. Types in simple theories have canonical bases, but the bases are represented as hyperimaginary elements instead of the imaginary elements used in the stable case. In some cases, such as supersimple or low simple theories, canonical bases can in fact be represented by imaginary elements. This behavior raises questions about how similar stable and simple theories are. There are obvious differences, such as the presence of unstable formulas in simple unstable theores, and the non-definability of types, since a type over a model in a simple theory can have unboundedly many global, non-forking extensions. Yet, because of the Type Amalgamation Theorem, extensions of that same type over a model to independent sets do have a common realization. Not 1 much is known about how a “generic” simple unstable theory differs from a stable theory. This work looks at three aspects of simple theories and uses them to give a better picture of the differences between the two classes. First, we introduce a property DWIP, and use it to show that a simple theory posesses it if and only if forking and thorn forking independence are equivalent and thorn forking has weak canonical bases. It is also shown that DWIP is equivalent to the weak elimination of hyperimaginaries. Second, we look at the stable forking conjecture, a strong statement asserting that simple unstable theories have an essentially stable “core”, and prove that it holds between elements having SU-rank 2 and finite SU-rank. Third, we consider a property on indiscernible sequences which is known to hold in every stable theory, and show it holds on, at most, a subset of simple theories out of all possible first order theories. The underlying concept among all these approaches is that of a canonical base of a type. In a stable theory, types have a unique smallest domain of definition, called the canonical base. The canonical base is not, in general, represented by elements in a structure, but rather by imaginary elements, That is, by names of classes of definable equivalence relations. In contrast, simple unstable theo- ries have canonical bases, but they cannot be represented as imaginary elements, instead hyperimaginary elements are needed. Much of the theory around hyper- imaginary elements was developed in order to provide a representation of canonical bases for forking-independence. The main issue with hyperimaginary elements is that they cannot be simply adjoined to a structure as imaginary elements can. Since a hyperimaginary element names a type-definable class, being able to name the class turns a closed set in the type space into a clopen set. Yet, some sub- 2 classes of simple theories have enough structure to allow canonical bases to be respresented by imaginary elements as in the stable case instead of hyperimagi- naries. This allows use to think of canonical bases as residing “in” the structure instead of beside it. One sufficient condition is for a theory has elimination of hyperimaginaries. The first comparison between stable and simple unstable theories comes from an early observation that although there are many examples of simple unstable theories, many of these seem to be essentially stable up to some kind of “noise.” So, in essence, there should not be much difference between the two classes, provided there is some way to define the word “noise,” and to account for it. A strong definition is to ask that every instance of forking in a simple unstable theory is always witnessed by a stable forking formula. The formalization of this is (see [10]) Definition 0.0.1. A simple theory T has stable forking if whenever q(x) is a complete type over a model M, A ⊂ M and q forks over A then there is is a stable formula ψ(x; b) ∈ q that forks over A. The Stable Forking Conjecture then asks whether every simple theory has stable forking. The conjecture is unresolved—and not universally believed true— but it has been verified in several special classes including 1-based supersimple theories and stable theories expanded with a generic predicate. There have been many attempts at showing pieces of it (see [9], [10], [13], and [15]). Some involve streghthing the conjecture, others look outside elementary first-order theories to homogenous models. Peretz considers the context of ω-categorical supersimple theories, and there shows that there is always stable forking between elements of rank 2, and that between elements of rank 3, any counterexample to the conjecture 3 must exhibit a very specific type of indiscernible sequence. We extend Peretz’s results by dropping the condition of ω-categoricity, and then showing stable forking between an elements of SU-rank 2 and of finite SU-rank, so long as the theory has weak-elimination of hyperimaginaries. To draw a second distinction between stable and simple, we look at a notion of independence a priori weaker than forking independence called thorn forking independence. Onshuus introduced thorn forking (see [11]) and showed it to be a useful notion of independence in a broad class of theories, christened rosy theories, which include all stable, simple, and o-minimal theories. He also showed thorn forking is equivalent to Shelah forking in stable theories [12]. However, the rela- tionship between forking and thorn forking in simple theories remained an open question. In [4] Ealy showed the two notions of independence are equivalent in simple theories having elimination of hyperimaginaries. The chief obstacle is that in a simple theory canonical bases are represented by hyperimaginary elements. In contrast, thorn forking is related to algebraic closure, and is intimately tied to the imaginary elements of a theory. In fact, thorn independence is sensitive to whether one is working in T or T eq. Elimination of hyperimaginaries allows canon- ical bases to be represented as imaginary elements, and thus allows thorn forking to respect forking (and thus, to coincide). We observe that weak elimination of hyperimaginaries, while not quite preserving canonical bases, preserves enough for thorn forking to detect their presence. A theory having weak elimination of hyperimaginaries is equivalent to the theory having the property dependence witnessed by imaginaries (DWIP), which asserts that all instances of forking are witnessed by imaginary elements. This permits a “transfer of dependence” from hyperimaginaries to imaginaries to get the connection between forking and thorn 4 forking. Finally, some classes of theories can be characterized by properties on indis- cernible sequences.
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