Simple Theories and Hyperimaginaries

Simple Theories and Hyperimaginaries In the 1990s Byunghan Kim and Anand Pillay generalized stability, a major modeltheoretic idea developed by Saharon Shelah twenty-five years earlier, to the study of simple theories. This book is an up-to-date introduction to simple theories and hyperimaginaries, with special attention to Lascar strong types and elimination of hyperimaginary problems. Assuming only knowledge of general model theory, the foundations of forking, stability, and simplicity are presented in full detail. The treatment of the topics is as general as possible, working with stable formulas and types and assuming stability of the theory only when necessary. The author offers an introduction to independence relations as well as a full account of canonical bases of types in stable and simple theories. In the last chapters, the notions of internality and analyzability are discussed and used to provide a self-contained proof of elimination of hyperimaginaries in supersimple theories. Enrique Casanovas is a Professor of Logic and Philosophy of Science in the Department of Logic, History and Philosophy of Science at the University of Barcelona.

lecture notes in logic

A Publication of The Association for Symbolic Logic This series serves researchers, teachers, and students in the field of symbolic logic, broadly interpreted. The aim of the series is to bring publications to the logic community with the least possible delay and to provide rapid dissemination of the latest research. Scientific quality is the overriding criterion by which submissions are evaluated. Editorial Board H. Dugald Macpherson, Managing Editor Department of Pure Mathematics, School of Mathematics, University of Leeds Jeremy Avigad Department of Philosophy, Carnegie Mellon University Vladimir Kanovei Institute for Information Transmission Problems, Moscow Manuel Lerman Department of Mathematics, University of Connecticut Heinrich Wansing Faculty of Philosophy and Educational Science, Ruhr-Universit¨at Bochum Thomas Wilke Institut fur Informatik, Christian-Albrechts-Universit¨at More information, including a list of the books in the series, can be found at http://www.aslonline.org/books-lnl.html.

Simple Theories and Hyperimaginaries ENRIQUE CASANOVAS Universidad de Barcelona

association for symbolic logic

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521119559  C Association of Symbolic Logic 2011

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Casanovas, Enrique, 1957– Simple theories and hyperimaginaries / Enrique Casanovas. p. cm. – (Lecture notes in logic) Includes bibliographical references and index. ISBN 978-0-521-11955-9 1. Model theory. 2. First-order logic. 3. Hyperspace. I. Title. II. Series. QA9.7.C37 2011 511.3 4 – dc22 2011005729 ISBN 978-0-521-11955-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Para Maribel

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2. ϕ-types, stability and simplicity . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 3.

∆-types and the local rank D(ð, ∆, k) . . . . . . . . . . . . . . . . . 19

Chapter 4. Forking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 5. Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 6. The local rank CB∆ (ð) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Chapter 7. Heirs and coheirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 8. Stable forking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 9. Lascar strong types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Chapter 10. The independence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Chapter 11. Canonical bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 12. Abstract independence relations . . . . . . . . . . . . . . . . . . . . . 75 Chapter 13. Supersimple theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 14. More ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Chapter 15. Hyperimaginaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter 16. Hyperimaginary forking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Chapter 17. Canonical bases revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Chapter 18. Elimination of hyperimaginaries . . . . . . . . . . . . . . . . . . . . . . 131 ix

x

Contents

Chapter 19. Orthogonality and analysability . . . . . . . . . . . . . . . . . . . . 143 Chapter 20. Hyperimaginaries in supersimple theories . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

PREFACE

These lecture notes originated in a seminar on Model Theory that I gave in the academic years 2005–06 and 2006–07 at the Department of Logic, History and Philosophy of Science of the University of Barcelona. I had presented some previous work on the basic notions of simple theories in July 2002 in the Simpleton Workshop held at the Centre International de Rencontres Math´ematiques, Luminy (Marseille), which was subsequently published as [8]. A more extended version, including the exposition of stable theories, was the topic of a tutorial entitled Advanced Stability Theory that I taught in the Modnet Summer School that took place at the University of Freiburg in April 2006. And in preparing the material, I also drew on some courses on these topics given at the Universidad de los Andes, Bogot´a, in August 2000 and in August 2004. The notes are based on the work of many model theorists. The names of John T. Baldwin, Ehud Hrushovski, Byunghan Kim, Daniel Lascar, Ludomir Newelski, Anand Pillay, Bruno Poizat, Saharon Shelah, Frank O. Wagner, and Martin Ziegler deserve special mention. I learned stability theory from Martin Ziegler and I have made as much use as I could of his short and elegant proofs, presented in his courses and in his unpublished lecture notes. This book is not as ambitious as Frank O. Wagner’s book on simple theories [41], but its pace might be more comfortable for the beginner. I have tried to fill in some gaps and give details of proofs, but this has meant that fewer topics can be covered. Even in the realm of pure simple theories there are many important areas that have not been discussed here. The aim was to present the foundations of simple theories, forking calculus, and the stable fragment of any theory with the greatest generality I could afford and, secondly, to develop the topics of Lascar strong types and hyperimaginaries exhaustively. My treatment of hyperimaginaries is complete up to two particular results: the theorem of Lascar–Pillay, based on Weil’s theorem on compact groups, stating that every bounded hyperimaginary is equivalent to a sequence of finitary hyperimaginaries, and the still unpublished result of Adler that in every theory with the strict order property there is an infinitary hyperimaginary that xi

xii

Preface

is not eliminable (see [26] for the first result and [1] for the second one). I have also omitted the discussion of hyperimaginaries in the context of abstract elementary classes and some recent results within the nonindependence property setting. Elimination of hyperimaginaries plays a fundamental role in the book. Let me summarize the situation. Stable and supersimple theories eliminate hyperimaginaries, low theories eliminate bounded hyperimaginaries, and small theories eliminate finitary hyperimaginaries; theories with the strict order property do not eliminate infinitary hyperimaginaries. Moreover, there is an example in [11] of a nonsimple theory without the strict order property that does not eliminate finitary hyperimaginaries. This is all we know. The ultimate goal is to lay out, with all its prerequisites, a fundamental theorem in Model Theory due to Buechler, Pillay, and Wagner: supersimple theories eliminate hyperimaginaries. The last two chapters of the book are devoted to presenting this result. In the Preliminaries chapter I fix notation and explain the framework, including the monster model, basic results on type-definability, imaginaries, and ˝ a very powerful result on indiscernibles based on the Erdos–Rado Theorem. I assume the reader is familiar with Model Theory. As basic reference texts I recommend the books of Hodges [13], Marker [28], Poizat [36], the lecture notes of Ziegler, and the recent books of Lascar [24], and of Tent and Ziegler [40]. Chapters 2–5 are devoted to the foundations of simple theories. In chapter 2 ϕ-types are discussed and the notions of ë-stability, stability, order property, independence property, strict order property, tree property, and simplicity are introduced. In chapter 3 local D-rank and dividing are defined and their basic properties are developed. In chapter 4 forking and dividing are discussed. In chapter 5 the ternary relation of independence based on nonforking is presented and, after giving a list of its more general properties that hold in any theory, we confine our study to the context of simple theories, proving local character, extension, symmetry, and transitivity properties. Morley sequences are also introduced and special attention is given to type-definability issues of nonforking and Morley sequences. In chapters 6–8 the stable fragment of any theory is studied. I adopt a local point of view, as in Pillay [32] and [33], focusing the study on the behavior of ϕ-types for stable formulas ϕ in arbitrary theories. There are two different notions of ϕ-formulas and ϕ-types in the literature. I reserve the denomination of ϕ-formula and ϕ-type for Shelah’s notion and I use the terms generalized ϕ-formula and generalized ϕ-type for the variants introduced by Pillay. The treatment of local Cantor–Bendixson rank in chapter 6 is based on Ziegler’s lecture notes [42]. This rank CB∆ (ð(x)) is not the same as Shelah’s local rank R(ð(x), ∆, ∞), unless ð(x) is a set of ∆-formulas. The presentation of the

Preface

xiii

topic of heirs and coheirs in chapter 7 is not complete since some interesting results of Lascar and Pillay (see [25]) have not been included. Chapter 9 is a general introduction to Lascar strong types, Kim–Pillaytypes, and strong types. Newelski’s Theorem on finiteness of diameter of type-definable Lascar strong types (Theorem 9.22 here) is proved following ideas from [30] as developed in Pel´aez’s doctoral dissertation [31]. Chapter 10 is devoted to the Independence Theorem for simple theories. First I prove the Independence Theorem for Lascar strong types (as Shami does in [37]) and then I derive from it the version for types over models. We take occasion to define G-compactness and present a few results about it. However, the Lascar group is not discussed here. In chapter 11 canonical bases of stationary types in simple theories are introduced. This is just the classical theory of canonical bases in stable theories, but slightly generalized to make it as well applicable in simple theories. The proper notion of canonical base for simple theories is postponed to chapter 17 since it requires the previous introduction of hyperimaginaries. Chapter 12 is an exposition of abstract independence relations, a topic of interest beyond simple theories. The treatment is axiomatic with special attention to nonforking and nondividing independence. Results from Adler’s two articles [2] and [3], such as the symmetry of any independence relation (Corol∗ lary 12.6 here) or the study of ^ | , play a fundamental role. The Kim–Pillay Theorem characterizing simple theories as theories with an independence relation satisfying the Independence Theorem over models is also proved, as well as similar characterizations of stable theories. Finally, Kim’s characterization (from [18]) of simple theories in terms of symmetry or transitivity of nonforking independence is given. Chapters 13 and 14 are an exposition of supersimple theories with special emphasis on ranks SU and D. There is also an axiomatic characterization of ranks useful for characterizing superstable theories. On this last point I follow Ziegler’s lecture notes. With chapter 15 begins the second part of the book, whose main point is to develop the model theory of hyperimaginaries. The main references are Hart, Kim, and Pillay’s seminal paper [12], Wagner’s book [41], and Lascar and Pillay’s article [26]. The first topics in chapter 15 are the equivalence of hyperimaginaries, types of hyperimaginaries, and the bounded closure operator. Then I study equality of Kim–Pillay-type relativized to a complete type and I give the proof that any bounded hyperimaginary is an equivalence class of a bounded type-definable equivalence relation (Proposition 15.27 here). In chapter 16 forking and independence are developed for hyperimaginaries. It is not a straightforward translation of the previously studied theory and very often the natural order of proving some results from previous ones changes dramatically. For instance, the local character of forking (Proposition 16.21

xiv

Preface

here) has a very different proof compared to the classical case. In a first step the Independence Theorem is proved for types over models and later for hyperimaginary Lascar strong types. We also pay attention to type-definability of nonforking and independence. The main application of hyperimaginaries is to provide canonical bases for amalgamation bases in simple theories. This is presented in chapter 17. In the first section of this chapter several ways of obtaining amalgamation bases in simple theories are examined carefully. Chapter 18 deals with the elimination of hyperimaginaries and related topics. After presenting some general results about elimination, the cases of stable theories, small theories and low theories are considered. The elimination of hyperimaginaries in supersimple theories is the topic of chapter 20. In chapter 19 all prerequisites for the elimination proof are developed in detail. This concerns mainly the study of the analysability rank Ran , which might be interesting in itself for further applications. Chapter 20 closes with Kim’s characterization of canonical bases in simple theories as sets of canonical parameters of definitions of p-stable formulas. We follow the short proof given by Kim and Pillay in [21]. Examples and commentaries are scarce throughout the book. My main concern has been with the compact presentation of the theory and the detailed exposition of proofs. I wish I had had the time and talent to embellish the text with funny comments and jokes, as the excellent writer Bruno Poizat does. Acknowledgments. I thank the participants in the Model Theory Seminar, Hans Adler, Silvia Barbina, Rafel Farr´e, Javier Moreno, Rodrigo Pel´aez, Juan Francisco Pons, and Joris Potier for their patience and their remarks. I thank Daniel Palac´ın and Joris Potier for their careful reading of previous versions. I also thank Anand Pillay and Frank Wagner for answering many questions I addressed to them concerning details of proofs of their results on the topics of this book. Finally, I thank an anonymous referee of the book for his detailed remarks and suggestions. Of course, any mistakes or inaccuracies that might still remain in the text are my sole responsibility.

Chapter 1

PRELIMINARIES

T is a complete theory of language L with infinite models and C is its monster model. Usually A, B, C are subsets of C and a, b, c are tuples of elements of C. We think of C as a proper class, so subset means small subclass. Some definitions also make sense for arbitrary subclasses A of C. For instance, for any subclass A of C, tp(a/A) is the class of all formulas with parameters in A realized by a in C, and Aut(C/A) is the group of all automorphisms of C which fix A pointwise. If A is a set, then tp(a/A) = tp(b/A) if and only if f(a) = b for some f ∈ Aut(C/A). Notation a ∈ A means that all the elements in the tuple a belong to A. We use x, y for single variables but also for tuples of variables. A tuple is a sequence, not necessarily finite. A class A of tuples is bounded if it is small compared with the monster model, hence if it is a set. The reader may feel more comfortable assuming the monster model is a saturated model of inaccessible cardinality κ; in this case he should replace the words proper class and set by of cardinality κ and of cardinality < κ respectively. Unless stated otherwise, κ, ë, ì are infinite cardinal numbers and α, â, ã are ordinal numbers. On is the class of all ordinal numbers. We use i, j, k for indexes, in most cases ordinal numbers. Sometimes it is convenient to add an end point ∞ to On. Hence we understand that ∞ = supα∈On α. The set of all mappings from I into J is denoted by J I . Hence, κ ë has two meanings: cardinal exponentiation and set of functions. The context will make clear which one is intended. For a sequence (ai : i < α) and any j ≤ α we use a<j for (ai : i < j). The union of two sets A, B is sometimes denoted by AB. A formula is over A if it has parameters in A. By ϕ(x) ∈ L(A) we mean that ϕ(x) is over A. In particular ϕ(x) ∈ L means that ϕ(x) is a formula without parameters. A partial type over A is a consistent set of formulas over A. A finitary type is a type in finitely many variables. For a formula ϕ(x) we write |= ϕ(a) or a |= ϕ(x) to mean C |= ϕ(a), and similarly for a type. The set of all complete types over A of a fixed length of tuples is denoted by S(A). This makes sense even when A is a proper subclass of C, but in this case the types are not necessarily realized in C. If we want to make clear that n or κ is the fixed length of the involved tuples we write Sn (A) or Sκ (A).

1

2

1. Preliminaries

S(A) is naturally a boolean topological space, that is, a compact Hausdorff totally disconnected topological space. A basis of clopen sets is given by [ϕ] = {p ∈ S(A) : ϕ ∈ p}. If Σ is a type over A, the corresponding closed set is [Σ] = {p ∈ S(A) : Σ ⊆ p}. We write a ≡A b to express equality of types over A: tp(a/A) = tp(b/A). For sets A, B we sometimes use notations like tp(A/C ) and A ≡C B. An implicit enumeration of A (and a corresponding one for B) should be assumed. Hence A ≡C B means that a ≡C b for some tuples a and b enumerating A and B respectively. Implication is denoted by Σ ` ϕ. If Σ is a set of formulas, it means that all realizations of Σ in the monster model also realize ϕ. If Σ is a complete type over A and ϕ ∈ L(A), this is obviously equivalent to ϕ ∈ Σ. If ð(x) is a partial type, ð(C) is the class of tuples (of the length of x) that satisfy ð, and more generally, ð(A) is the set of all tuples a ∈ A such that |= ð(a). Equivalence of partial types Σ(x) and ð(x) (perhaps over different sets) means Σ(C) = ð(C). We write Σ(x) ≡ ð(x) for it. Of course, it can be rephrased as Σ ` ϕ for all ϕ ∈ ð and ð ` ϕ for all ϕ ∈ Σ. A model M is always S an elementary submodel of the monster model C. In fact, C is the union i∈On Mi of an elementary chain (Mi : i ∈ On) of models, where each Mi is |i|-saturated. Very often we say a proof or a step in a proof is by compactness. Sometimes this means that if Σ ` ϕ then Σ0 ` ϕ for some finite Σ0 ⊆ Σ, but sometimes it only means we are claiming some set of formulas is consistent and we are using the fact that the monster model C is saturated: every type over a (small) subset is realized in C. Of course, the consistency of the involved set of formulas should be checked. When we say that a set of formulas Σ(x) is consistent or inconsistent we really mean consistency or inconsistency relative to our theory T . If Σ(x) is over A it should be understood that we mean relative to the theory T (A) of the monster model expanded by constants for the elements of A. This amounts to being the same for Σ as being finitely satisfiable in the monster model (expanded by the parameters in A). It is convenient to work with complete types over the monster model. We call them global types. It makes sense to consider the space of types S(C) although every global type p ∈ S(C) is in fact a proper class. It is similar to what happens with the group Aut(C). The background set theory needed to deal with this situation is a finite iteration of the Bernays–von Neumann– ¨ Godel theory of classes (with choice), which is a conservative extension of ZFC . We often consider properties of formulas and discuss them. In fact they are usually properties of a formula ϕ together with a separation of disjoint tuples of variables x, y including all the variables occurring in the formula. The same formula with a different choice of tuples might not have the property.

1. Preliminaries

3

The reader should notice that when we say ϕ(x, y) has a certain property we really mean that the formula has the property with respect to the displayed separation of variables x, y. When several tuples appear and we want to make clear which is the intended separation, we use notations like ϕ(x; y, z) or ϕ(x, y; z). A particular case of all this appears when we start with a formula ϕ(x, y) and we want to consider the same formula but with the opposite separation of tuples of variables: y, x. In this case we use ϕ −1 (y, x) for the new situation. Hence |= ϕ(a, b) ⇔ |= ϕ −1 (b, a). The group Aut(C/A) acts naturally on C but also on the space of types S(C). The image of an object by some f ∈ Aut(C/A) is often called an A-conjugate of the object by f. If ϕ(x, y) ∈ L, the A-conjugate of ϕ(x, a) by f ∈ Aut(C/A) is the formula ϕ(x, a)f = ϕ(x, f(a)). Similarly for types. If p ∈ S(C), the A-conjugate of p by f ∈ Aut(C/A) is the global type pf ∈ S(C) all whose formulas are conjugate by f of formulas in p. Hence pf = {ϕ(x)f : ϕ(x) ∈ p}. Sometimes it is convenient to look at Aut(C/A) as a topological group. The natural topology on Aut(C/A) is the topology of pointwise convergence. A basis of open sets is given by {f : f(a) = b} for any finite tuples a, b ∈ C. An element a ∈ C is definable over the set Aif its orbit under Aut(C/A) is a singleton and it is algebraic over Aif the orbit is finite. For any set A, the set of all elements definable over A is the definable closure of A, denoted dcl(A), and the set of all elements algebraic over A is the algebraic closure of A, denoted acl(A). These definitions can be extended to the case of an arbitrary class A ⊆ C by letting dcl(A) be the union of all dcl(B) where B ranges over subsets of A, and similarly for acl(A). Notice that dcl and acl are finitary closure operators on subclasses of the monster model, i.e., they are operators cl such that 1. 2. 3. 4.

A ⊆ cl(A). If A ⊆ B, then cl(A) ⊆ cl(B). cl(cl(A)) ⊆ cl(A). If a ∈ cl(A), then a ∈ cl(A0 ) for some finite A0 ⊆ A.

A partial type ð(x) over A is called algebraic (over A) if all realizations of ð belong to acl(A). Definition 1.1. A relation R (a subclass of CI for some index set I ) is Ainvariant if it is preserved under automorphisms of C fixing A pointwise, that is, R(a) ⇒ R(f(a)) for all a, for all f ∈ Aut(C/A). This clearly implies R(a) ⇔ R(f(a)). Remark 1.2. For every set A, every A-invariant relation W VR is definable by a disjunction of types over A, namely: R(a) ⇔ a |= R(b) tp(b/A).

4

1. Preliminaries

Definition 1.3. A type-definable over Arelation R is a relation for which there is a set of formulas ð(x) over A such that ð(C) = R, that is, R(a) ⇔ |= ð(a) for all a. If A = ∅ we say that R is 0-type-definable. As usual, if ð is in fact a formula we talk of definability over A or A-definability. A 0-definablerelation is a relation definable without parameters. Lemma 1.4. 1. If R is type-definable over A and it is B-invariant, then it is type-definable over B. The same is true for definability. 2. If R and the complement R¯ of R are type-definable over A, then R is definable over A. 3. The class of all type-definable over A relations is closed under arbitrary intersections, finite unions, and quantification of any number of variables. Proof. 1. Let ð(x) be a type over A defining R and consider the restriction map f : S(AB) → S(B) given by f(p) = p  B. It is a continuous function from a compact space onto a Hausdorff space, and therefore it is closed. The image of the closed set [ð] = {p ∈ S(AB) : ð ⊆ p} in f is closed and therefore it is of the form [Σ] for some partial type Σ over B. By B-invariance it is easy to check that Σ(C) = R. 2. Let ð1 (x), ð2 (x) be sets of formulas over A defining R and its complement respectively. Since ð1 (x) ∪ ð2 (x) is inconsistent, there is a finite conjunction ϕ(x) of formulas of ð1 (x) which is inconsistent with ð2 (x). Clearly, ϕ(x) defines R. 3. For instance, if Σ(x, y) defines R and it is closed under finite conjunctions, then {∃yϕ(x, y) : ϕ(x, y) ∈ Σ} defines {a : ∃yR(a, y)}. a Definition 1.5. Let (I, <) be a linearly ordered set and let a = (ai : i ∈ I ) be a sequence of tuples ai of the same length. We say that a is A-indiscernible or is indiscernible over Aif for every n < ù, whenever i0 < · · · < in and j0 < · · · < jn are indexes in I , then ai0 , . . . , ain ≡A aj0 , . . . , ajn . The existence of indiscernible sequences is usually established using Ramsey’s Theorem. Here it is convenient to introduce a more powerful method ˝ based on the Erdos–Rado Theorem. Proposition 1.6. If κ ≥ |T | + |A|, ë = i(2κ )+ , and (ai : i < ë) is a sequence of tuples ai of the same length ≤ κ, then there is an A-indiscernible sequence (bi : i < ù) such that for each n < ù there are i0 < · · · < in < ë such that b0 , . . . , bn ≡A ai0 , . . . , ain . S Proof. (bi : i < ù) is a realization of a type p = p(xi : i < ù) = n<ù pn (x1 , . . . , xn ) where pn ∈ Sn (A) and pn ⊆ pn+1 . The types pn are constructed inductively together with a descending chain (In : n < ù) of subsets of (2κ )+ of cardinality |In | = (2κ )+ and a family (Xin : i ∈ In ) of subsets Xin of ë in such a way that 1. Xin+1 ⊆ Xin for all i ∈ In+1 .

1. Preliminaries

5

2. |Xin | > i2κ +α if i is the α-th element of In . 3. (aj1 , . . . , ajn ) |= pn for all j1 < · · · < jn ∈ Xin for all i ∈ In . We start with I0 = (2κ )+ , Xi0 = ë, and p0 = ∅. Let (îα : α < (2κ )+ ) enumerate increasingly In , and set In0 = {îα+n : α < (2κ )+ }. Let i ∈ In0 , say i = îα+n where α < (2κ )+ . There are at most 2κ types tp(aj1 , . . . , ajn+1 /A) ˝ for j1 < · · · < jn+1 ∈ Xin . Since |Xin | > i2κ +α+n = in (i2κ +α ), by the Erdos– n+1 + + n+1 n Rado Theorem ( (in (κ)) → (κ )κ ) there is some Xi ⊆ Xi and some i pn+1 ∈ Sn+1 (A) such that |Xin | > i2κ +α and for all j1 < · · · < jn+1 ∈ Xin+1 , i (aj1 , . . . , ajn+1 ) |= pn+1 . Since |In0 | = (2κ )+ we can choose In+1 ⊆ In0 such that κ + i i0 |In+1 | = (2 ) and pn+1 = pn+1 = pn+1 for all i, i 0 ∈ In+1 . a Corollary 1.7. 1. If (ai : i < ë) is indiscernible over A, there is some model M ⊇ A such that (ai : i < ë) is indiscernible over M . 2. If (ai : i < ë) is indiscernible over A, then it is also indiscernible over acl(A). Proof. 1. Choose a model M ⊇ A and extend the sequence to an Aindiscernible sequence (ai : i < ë0 ) large enough to apply Proposition 1.6. This gives an M -indiscernible sequence (bi : i < ù) such that (bi : i < ù) ≡A (ai : i < ù). Now extend this sequence to an M -indiscernible sequence (bi : i < ë) and choose N such that (ai : i < ë)N ≡A (bi : i < ë)M . Then (ai : i < ë) is indiscernible over the model N ⊇ A. 2. By 1, since acl(A) ⊆ M for any model M ⊇ A. a The many-sorted structure M eq is constructed from M |= T , adding a new sort M n /E for each 0-definable equivalence relation E on a finite power M n of M and extending the language L to Leq adding a corresponding n-ary function symbol ðE for the projection map, the map ðE : M n → M n /E which sends each a ∈ M n to its equivalence class ðE (a) = aE . We identify M with M/ =, and we call it the home sort. Note that all symbols of L are in Leq and the home sort M carries its original structure. In fact the construction should be described syntactically, starting with the formula defining E instead of dealing with E itself. It is a uniform procedure in every model of T and everything applies in particular to the monster model C of T . The corresponding theory T eq is, by definition, the complete theory of Ceq . Elements of Ceq are called imaginaries. We call elements of C real elements. Sometimes it is convenient to consider imaginaries in T (A). They are equivalence classes of tuples in A-definable equivalence relations and they are called A-imaginaries. We define all notions for the one-sorted theory T , but it is a very instructive exercise to adapt the definitions to the many-sorted case and to check that the corresponding notions have the same properties in T eq . This is not always trivial. Every general result we prove for a theory T can also be applied to T eq , only taking care of the right translation to the many sorted framework.

6

1. Preliminaries

Lemma 1.8. For every formula ϕ(y, x1E1 , . . . , xnEn ) ∈ Leq , where y is a tuple of variables of the home sort and each xiEi is a variable of sort Ei (an equivalence relation on mi -tuples), there is a formula ø(y, y1 , . . . , yn ) ∈ L, where each yi is a new mi -tuple of variables, such that for all tuples a, a1 , . . . , an ∈ M of the corresponding length, M eq |= ϕ(a, ðE1 (a1 ), . . . , ðEn (an )) ⇔ M |= ø(a, a1 , . . . , an ). Proof. In the first step we replace all quantifiers by quantification over the home sort using new variables. For example, we replace ∃xiEi by ∃yi and then we replace every occurrence of xiEi by ðEi (yi ). In the second step the projections ðEi are eliminated: ðEi (x) = ðEi (x 0 ) is replaced by Ei (x, x 0 ). a Notations like dcleq (A) and acleq (A) have the obvious meaning: definable and algebraic closure computed in Ceq . Proposition 1.9. 1. Ceq is the monster model of T eq . 2. M eq  Ceq if we identify each equivalence class aE in M with the corresponding equivalence class in C. 3. Every automorphism of M extends uniquely to an automorphism of M eq and we can identify Aut(M ) with Aut(M eq ). 4. M eq = dcleq (M ). 5. dcleq (A) ∩ C = dcl(A) and acleq (A) ∩ C = acl(A) for every A ⊆ C. 6. Every relation definable in M eq with parameters is definable in M eq with parameters of M . 7. Every R ⊆ M n definable in M eq over A ⊆ M is definable over A in M . Proof. Easy exercise. a A canonical parameter of a definable relation R is an imaginary element c such that for all f ∈ Aut(C), f(R) = R if and only if f(c) = c. It is unique up to interdefinability and it can be constructed in Ceq as follows: take some ϕ(x, y) ∈ L and some tuple of parameters a ∈ C such that ϕ(C, a) = R, define an equivalence relation E by E(b, d ) ⇔ ϕ(C, b) = ϕ(C, d ) and finally put c = aE . Lemma 1.10. Every A-imaginary is interdefinable with some imaginary. Proof. If e = aE is an A-imaginary, the canonical parameter e 0 of the equivalence class aE considered as a definable relation is interdefinable with e. a An equivalence relation is called finiteif it has only finitely many classes. Proposition 1.11. The following are equivalent for any definable relation R ⊆ Cn and any set A ⊆ C: 1. R is definable over any model M ⊇ A. 2. R has only finitely many A-conjugates.

1. Preliminaries

7

3. R is a union of equivalence classes of some finite A-definable equivalence relation. 4. R is definable over acleq (A). Proof. 1 ⇒ 2. Assume R has infinitely many conjugates over A and choose a model M ⊇ A, say of cardinality κ ≥ |T |. By compactness, R has at least κ + conjugates over A and they are all definable over M , which is impossible. V 2 ⇒ 3. Let R1 , . . . , Rm be a list of all A-conjugates of R. Define E(x, y) ⇔ 1≤i≤m (Ri (x) ↔ Ri (y)). E is a finite A-definable equivalence relation and R is a union of E-classes. 3 ⇒ 4. Assume E is a finite A-definable equivalence relation and R is the union of the classes [a1 ]E , . . . , [am ]E . By Lemma 1.10 each [ai ]E is interdefinable with some imaginary ei . Then ei ∈ acleq (A) and R is definable over e1 , . . . , em . 4 ⇒ 1. Clear since acleq (A) ⊆ M eq if M ⊇ A. a Iteration of the process T 7→ T eq is useless since (T eq )eq can be identified with T eq . In particular each definable relation in Ceq has a canonical parameter in Ceq . Corollary 1.12. If A is a finite set of imaginaries, there exists an imaginary c such that for all f ∈ Aut(C), f fixes setwise A if and only if f(c) = c. Proof. Since each finite tuple of imaginaries is interdefinable with a single imaginary, we may assume that all imaginaries in A are of the same sort. Hence A is a definable relation in Ceq and has a canonical parameter c ∈ Ceq . a The Cantor–Bendixson rank of points and closed sets in a topological boolean space X plays a key role in the presentation of stable theories and it is also relevant for small theories. We give here a brief summary of the results that will be needed later. In the applications sometimes X will be S(C) or Sϕ (C), which is a proper class. What we present here is also valid for this situation, with the only possible exception of Proposition 1.18. Definition 1.13. Let X be a topological space. For any A ⊆ X , let A0 be the set of all accumulation points of A. If A is closed, then A0 is a closed subset of A. For any set A, the topological closure of A in X is A ∪ A0 . The Cantor–Bendixson derivative of the space X is defined inductively as follows: 1. X (0) = X . (α) 0 2. X (α+1) = ). T (X (i) (α) 3. X = i<α X if α is a limit ordinal number. S 4. X (∞) = α∈On X (α) . Notice that each X (α) is closed and X = X (0) ⊇ . . . X (α) ⊇ X (α+1) ⊇ · · · ⊇ X (∞) . Remark 1.14. Let X be a compact topological space. 1. If α is a limit ordinal and X (i) 6= ∅ for all i < α, then X (α) 6= ∅.

8

1. Preliminaries 2. If α is the largest ordinal such that X (α) 6= ∅, then X (α) is finite. Proof. By compactness of X .

a

Definition 1.15. Let X be a compact topological space. The Cantor– Bendixson rank of a point a ∈ X , CBX (a), is ∞ if a ∈ X (∞) and otherwise it is the largest ordinal α such that a ∈ X (α) . If A ⊆ X is a nonempty closed set, we set CBX (A) = ∞ if there is some point a ∈ A with CBX (a) = ∞ and otherwise CBX (A) is the maximal ordinal α for which there is some a ∈ A with CBX (a) = α. In the first case the Cantor–Bendixson degree of A is ∞ and in the second case it is the number n < ù of points a ∈ X with CBX (a) = α. If A = ∅ we agree that CBX (A) = −1 and its Cantor–Bendixson degree is 0. The space X is scattered if X (∞) = ∅, that is, if every a ∈ X has ordinal Cantor–Bendixson rank CBX (a) < ∞. Recall that a boolean space is a compact Hausdorff space which is totally disconnected, i.e., which has a basis of clopen sets. Proposition 1.16. Let X be a boolean topological space and assume A ⊆ X is closed. There is some clopen set U ⊇ A with the same Cantor–Bendixson rank and degree as A. Hence, CBX (A) = min{CBX (U ) : U ⊇ A is clopen}. Proof. This is clear if A = ∅ or CBX (A) = ∞. Assume 0 ≤ α = CBX (A) < ∞. Then A ∩ X (α+1) = ∅ and for each a ∈ A there is some clopen set Ua such that a ∈ Ua and Ua ∩ X (α+1) = ∅. By compactness finitely many sets Ua cover A and their union is a clopen set U ⊇ A such that U ∩ X (α+1) = ∅. There are only finitely many points a ∈ U r A with CBX (a) = α and for each one we can find a clopen set Va such that a ∈ Va and Va ⊆ U r A. If V is the union of these clopen sets Va , then U r V is a clopen set extending A with the same Cantor–Bendixson rank and degree as A. a Proposition 1.17. In any boolean topological space X , we can compute the Cantor–Bendixson rank of a clopen set U ⊆ X according to the following rules: 1. CBX (U ) ≥ 0 if and only if U 6= ∅. 2. CBX (U ) ≥ α + 1 if and only if there is a sequence (Un : n < ù) of pairwise disjoint clopen subsets Un ⊆ U such that CBX (Un ) ≥ α for all n < ù. 3. CBX (U ) ≥ α for a limit ordinal α if and only if CBX (U ) ≥ i for all i < α. 4. If CBX (U ) = α < ∞, then the Cantor–Bendixson degree of U is the largest n < ù for which there are pairwise disjoint clopen subsets U1 , . . . , Un of U such that CBX (Ui ) ≥ α for all i = 1, . . . , n. Moreover, the Cantor–Bendixson rank of a point a ∈ X can be computed in terms of clopen sets as follows: CBX (a) = min{CBX (U ) : U is clopen and a ∈ U ⊆ X }.

1. Preliminaries

9

Proof. Left to the reader. a Proposition 1.18. Let X be a boolean topological space and assume U ⊆ X is clopen. Then CBX (U ) = ∞ if and only if there is a binary tree (Us : s ∈ 2<ù ) of nonempty clopen subsets Us ⊆ U such that Us is the disjoint union of Us a 0 and Us a 1 for all s ∈ 2<ù . Proof. ⇒. Since X is a set, there is some ordinal α such that X (α) = X (α+1) . Then CBX (a) = ∞ if and only if CBX (a) ≥ α for all points a ∈ X . Assuming CBX (U ) = ∞, we start the construction of the tree with U∅ = U . Assuming inductively CBX (Us ) = ∞ ≥ α + 1 we can split Us into two disjoint clopen sets Us a 0 and Us a 1 of rank ≥ α and hence of rank = ∞. ⇐. Assume 0 ≤ CBX (U ) < ∞ and choose Us of minimal ordinal rank α and of minimal degree among the clopen sets in the tree of rank α. Since Us is the disjoint union of Us a 0 and Us a 1 , one of them must have smaller rank or the same rank and smaller degree, a contradiction. a Corollary 1.19. If the boolean topological space X has smaller cardinality than the continuum 2ù , then it is scattered, i.e., CBX (a) < ∞ for all a ∈ X . Proof. By Proposition 1.18. a Proposition 1.20. If the boolean topological space X has a countable basis of open sets and it is scattered, then it is countable. Proof. Fix a countable basis {On : n < ù} of clopen sets and choose for each point p ∈ X with CBX (p) = α some n < ù such that On isolates p in X (α) . It is easy to see that the assignment is one-to-one. a

Chapter 2

ϕ-TYPES, STABILITY AND SIMPLICITY

Definition 2.1. Let ϕ(x, y) ∈ L. A ϕ-formula over A is a formula of the form ϕ(x, a) or ¬ϕ(x, a) with a ∈ A. A ϕ-type over A is a consistent set of ϕ-formulas over A. A ϕ-type p(x) over A is complete if for every a ∈ A either ϕ(x, a) ∈ p or ¬ϕ(x, a) ∈ p. The set of all complete ϕ-types over A is Sϕ (A). If p(x) is a complete type over A, the restriction p  ϕ is defined as the set of all ϕ-formulas of p. Clearly, p  ϕ ∈ Sϕ (A). This terminology and notation also apply to global types. Remark 2.2. One might also define ϕ-types over A as consistent sets of boolean combinations of ϕ-formulas over A. Every complete ϕ-type in our sense can be extended uniquely to a complete ϕ-type in this apparently wider sense. Hence, nothing new is really gained. But in the literature there is also a truly different notion of ϕ-formula over A. It is defined as a formula over A which is equivalent to a boolean combination of formulas of the form ϕ(x, a) where a is not necessarily a tuple of A. See Definition 6.9. Definition 2.3. Let ϕ(x, y) ∈ L. A complete ϕ-type p(x) over B is definable over A or A-definable if there is a formula ø(y) ∈ L(A), called a definition of p, such that for all a ∈ B, ϕ(x, a) ∈ p ⇔ |= ø(a). A standard notation for the defining formula ø(y) is dp xϕ(x, y). If A is not mentioned, we understand that A = B. A complete type p(x) ∈ S(B) is definable (over A) if all its restrictions p  ϕ are definable (over A). Lemma 2.4. Let p(x) ∈ Sϕ (M ) be definable and let B ⊇ M . 1. There is some q(x) ∈ Sϕ (B) extending p which is definable over M . 2. If q1 (x), q2 (x) ∈ Sϕ (B) are M -definable extensions of p, then q1 = q2 . Proof. 1. Let ø(y) ∈ L(M ) be a definition of p and set q(x) = {ϕ(x, a) : a ∈ B, |= ø(a)} ∪ {¬ϕ(x, a) : a ∈ B, |= ¬ø(a)}. 11

12

2. ϕ-types, stability and simplicity

The completeness of q is clear, and its consistency follows from the fact that M (and hence also C) satisfies the sentences ^ ^ ∀x1 . . . xn y1 . . . yn ( ø(xi ) ∧ ¬ø(yi ) → ∃x( ϕ(x, xi ) ∧ ¬ϕ(x, yi ))). 1≤i≤n

1≤i≤n

2. If q1 , q2 ∈ Sϕ (B) are M -definable extensions of p with definitions ø1 (y), ø2 (y) ∈ L(M ), then M |= ∀y(ø1 (y) ↔ ø2 (y)), which implies C |= ∀y(ø1 (y) ↔ ø2 (y)) and therefore q1 = q2 . a Definition 2.5. Let ë be an infinite cardinal number. We say that ϕ = ϕ(x, y) ∈ L is ë-stable or stable in ë if for any set A, |A| ≤ ë ⇒ |Sϕ (A)| ≤ ë. It is said that ϕ is stable if it is stable in some ë. Otherwise ϕ is called unstable. Proposition 2.6. The following conditions are equivalent for any formula ϕ = ϕ(x, y) ∈ L. 1. ϕ(x, y) is stable. 2. Γϕ (ù) is inconsistent, where for any ordinal α, Γϕ (α) = {ϕ(xf , yfi )f(i) : f ∈ 2α , i < α} and where ϕ 0 = ϕ and ϕ 1 = ¬ϕ. 3. For any set A, any type p(x) ∈ Sϕ (A) is definable. 4. ϕ(x, y) is ë-stable for all ë. Moreover in 3. one can add that p is definable by a formula of the form ø(y) = ∃x1 . . . xn ∃y1 . . . ym ÷(y, x1 , . . . , xn , y1 , . . . , ym ) where ÷ is a conjunction of formulas of the form ϕ(xi , yj ), ¬ϕ(xi , yj ), ϕ(xi , y), and ϕ(xi , y)-formulas over A. Proof. 1 ⇒ 2. Assume Γϕ (ù) is consistent. Let ë be an infinite cardinal number and let ì be the least cardinal number such that 2ì > ë. Then 2<ì ≤ ë. Since Γϕ (ì) is also consistent, there is a tree (bs : s ∈ 2<ì ) such that for every f ∈ 2ì the set of ϕ-formulas pf (x) = {ϕ(x, bfi )f(i) : i < ì} is consistent. Since pf (x) is inconsistent with pf 0 (x) whenever f 6= f 0 , it follows that there are 2ì > ë complete ϕ-types over the set A = {bs : s ∈ 2<ì }. Since |A| ≤ ë, this shows that ϕ(x, y) is not ë-stable. S 2 ⇒ 3. Let p(x) ∈ Sϕ (A) and assume Γϕ (ù) is inconsistent. Then Γϕ (ù) ∪ f∈2ù p(xf ) also is inconsistent and by compactness there is a least natural number n for which [ Γϕ (n) ∪ p(xf ) f∈2n

2. ϕ-types, stability and simplicity

13

is inconsistent. Again S by compactness, there is a finite subset p0 (x) ⊆ p(x) such that Γϕ (n)∪ f∈2n p0 (xf ) is inconsistent. Then n > 0 and one can check that for any a ∈ A, [ ϕ(x, a) ∈ p ⇔ Γϕ (n − 1) ∪ p0 (xf ) ∪ {ϕ(xf , a)} is consistent, f∈2n−1

that is: ϕ(x, a) ∈ p if and only if ^ ^ |= ∃(xf : f ∈ 2n−1 )∃(ys : s ∈ 2
which is a definition of p of the form indicated above. 3 ⇒ 4. Since there are at most ë many definitions of the described form over a set A with |A| ≤ ë, there are also ≤ ë many complete ϕ-types over A. This shows that ϕ(x, y) is ë-stable for any ë, but we are using not only hypothesis 3 but also the added information on the form of the definition. Without this extra information we can only guarantee that ϕ is stable in every ë ≥ |T |. But in fact this is enough to prove 3 ⇒ 1 and after all, using the additional information on the form of the definition, we have established that 1 implies 4. a Remark 2.7. 1. If ë ≥ |T | and all complete ϕ-types over models of cardinality ë are definable, then ϕ is ë-stable. 2. If ϕ is stable, then any global ϕ-type p ∈ Sϕ (C) is definable. Proof. 1. If ë ≥ |T |, any parameter set A of cardinality ≤ ë can be extended to a model M of cardinality ë. Therefore |Sϕ (A)| ≤ |Sϕ (M )| ≤ ë, since there are at most ë definitions of ϕ-types over M . 2. The proof of 2 ⇒ 3 given for Proposition 2.6 works also for p ∈ Sϕ (C). a Definition 2.8. ϕ = ϕ(x, y) ∈ L has the order property if there are (ai : i < ù), (bi : i < ù), such that: |= ϕ(ai , bj ) ⇔ i < j. Remark 2.9. 1. ϕ(x, y) has the order property if and only if there are (ai : i < ù), (bi : i < ù), such that |= ϕ(ai , bj ) ⇔ i ≤ j. 2. In the definition of the order property one can change the index set ù and its order by any infinite linear ordering. 3. ϕ(x, y) has the order property if and only if ¬ϕ(x, y) has the order property. 4. ϕ(x, y) has the order property if and only if ϕ −1 (y, x) has the order property. Proof. 1. Replace bj by bj0 = bj+1 . For 2 apply compactness. 3. First use 2 with the reverse order ù ∗ of ù, and then replace the strict order by the corresponding reflexive one as in 1. Finally 4 follows from 2 using ù ∗ again. a

14

2. ϕ-types, stability and simplicity

Lemma 2.10. Assume ϕ = ϕ(x, y) does not have the order property. If p(x) ∈ Sϕ (A) is finitely satisfiable in A (which is always true if A is a model ), then p is definable by a positive boolean combination of formulas of the form ϕ(a, y) with a ∈ A. Proof. Let X1 , . . . , Xn be a family of subsets of a set A. Consider the relation R(a, b) between elements a, b of A defined by R(a, b) ⇔ ∀i(1 ≤ i ≤ n)(a ∈ Xi → b ∈ Xi ). It is easy to see that a subset X ⊆ A is a positive boolean combination of the sets X1 , . . . , Xn if and only if b ∈ X whenever a ∈ X and R(a, b). The reason is that in this situation _^ X (x) ⇔ {Xi (x) : a ∈ Xi }. a∈X

We will use this result. Assume p is not definable by a positive boolean combination of formulas of the described form. We inductively choose tuples ai , bi , ci (i < ù) of elements of A. Suppose aj , bj , cj are defined for all j < i. By hypothesis, {a ∈ A : ϕ(x, a) ∈ p} is not a positive boolean combination of the sets Xj = {a ∈ A :|= ϕ(cj , a)} for j < i. Then there are ai , bi ∈ A such that ϕ(x, ai ) ∈ p, ¬ϕ(x, bi ) ∈ p and for all j < i, if |= ϕ(cj , ai ), then |= ϕ(cj , bi ). Now let ci be a realization in A of the finite type p  {aj , bj : j ≤ i}. The sequences of tuples obtained this way have the property that |= ϕ(cj , ai ) ∧ ¬ϕ(cj , bi ) for i ≤ j but |= ϕ(cj , ai ) → ϕ(cj , bi ) for j < i. By Ramsey’s Theorem we may assume that always |= ¬ϕ(cj , ai ) for j < i or always |= ϕ(cj , bi ) for j < i. In the first case we have i ≤ j if and only if |= ϕ(cj , ai ). In the second case i ≤ j if and only if |= ¬ϕ(cj , bi ). In either case ϕ(x, y) has the order property. a Proposition 2.11. ϕ(x, y) is stable if and only if it does not have the order property. Proof. If ϕ(x, y) has the order property, then there are ai , bj (i, j ∈ Q) such that for all i, j |= ϕ(ai , bj ) ⇔ i < j. Now for each real number r, let pr (x) be the ϕ-type {ϕ(x, bj ) : r < j} ∪ {¬ϕ(x, bj ) : r ≥ j}. Clearly pr (x) is inconsistent with ps (x) if r 6= s and thus there are 2ù complete ϕ-types over the countable set {bj : j ∈ Q}. Hence ϕ is not stable. The other direction follows from Lemma 2.10 and the first point in Remark 2.7. a Corollary 2.12. Any boolean combination ϕ(x1 , . . . , xn ; y1 , . . . , yn ) of stable formulas ϕi (xi , yi ) is stable. The tuples xi and xj (and also yi and yj ) may have elements in common, but xi and yj are assumed to be disjoint.

2. ϕ-types, stability and simplicity

15

Proof. Without loss of generality xi = xj = x and yi = yj = y. For the case ¬ϕ(x, y) use Remark 2.9 and for ϕ(x, y) ∨ ø(x, y) use Ramsey’s Theorem. a Remark 2.13. Let ϕ = ϕ(x, y) ∈ L be stable. By Remark 2.7, any p(x) ∈ Sϕ (C) is definable over some set A. If M ⊇ A, then p(x) has a definition which is a positive boolean combination of formulas of the form ϕ(a, y) with a ∈ M and which is, at the same time, equivalent to a formula over A. Proof. Since ϕ does not have the order property, we can use Lemma 2.10, which gives a positive boolean combination ø(y) ∈ L(M ) of formulas of the form ϕ(a, y) which defines p  M . Since there is only one global ϕ-type extending p  M and definable over M , and ø defines in C a ϕ-type with these properties, it follows that ø defines p and hence ø is equivalent to a formula over A. a Definition 2.14. ϕ(x, y) ∈ L has the strict order property if there is a sequence (ai : i < ù) such that for all i < j < ù, ϕ(C, ai ) ( ϕ(C, aj ). Remark 2.15. 1. Clearly, a formula with the strict order property has the order property. 2. In the definition of the strict order property one can replace the ordered set (ù, <) by any other infinite linearly ordered set. 3. If the formula ϕ(x, y, a) has the strict order property, then also ϕ(x; y, z) has the strict order property. 4. There is a formula in T with the strict order property if and only if for some n there is a 0-definable partial order of Cn which has infinite chains. In fact if ϕ(x, y) has the strict order property, then ø(y1 , y2 ) = ∀x(ϕ(x, y1 ) → ϕ(x, y2 )) ∧ ∃x(ϕ(x, y2 ) ∧ ¬ϕ(x, y1 )) defines such an order. Proof. It is an easy exercise.

a

Definition 2.16. ϕ(x, y) ∈ L has the independence property if there are sequences (ai : i < ù) and (bX : X ⊆ ù) such that for all i, X , |= ϕ(ai , bX ) ⇔ i ∈ X. Remark 2.17. 1. A formula ϕ(x, y) with the independence property is unstable. In fact, for every cardinal ë there is some set A of cardinality ë such that |Sϕ (A)| = 2ë . 2. ϕ(x, y) has the independence property if and only if for each n < ù there is a sequence (ai : i < n) such that for each X ⊆ n {ϕ(ai , y) : i ∈ X } ∪ {¬ϕ(ai , y) : i ∈ n r X } is consistent.

16

2. ϕ-types, stability and simplicity

3. If ϕ(x, y) has the independence property, then also ϕ −1 (y, x) has the independence property. Proof. 1 and 2 are compactness arguments. For 3, for any n < ù, there are (aX : X ∈ P(n)), (bI : I ⊆ P(n)) such that |= ϕ(aX , bI ) ⇔ X ∈ I . Let Ui = {X ⊆ n : i ∈ X } for i < n and let ci = bUi . Then |= ϕ −1 (ci , aX ) ⇔ i ∈ X. a Proposition 2.18. There is an unstable formula in T if and only if there is a formula with the strict order property or there is a formula with the independence property. In fact if ϕ(x, y) is an unstable formula without the independence property, then the following holds: 1. Some conjunction of ϕ(x, y) with formulas of the form ϕ(x, a) and ¬ϕ(x, a) has the strict order property. 2. For ù, for some s ∈ 2n , the formula ø(x; y1 , . . . , yn ) = V some n < s(i) has the strict order property (where ϕ 0 = ϕ and i
i∈nrS

S is not an initial segment of n because otherwise some aj would represent it. But S is obtained as the last step S = Sm in a sequence S0 , . . . , Sm of subsets Sk ⊆ n where S0 is an initial segment and for each k there is some m ∈ Sk such that m + 1 6∈ Sk and Sk+1 = (Sk r {m}) ∪ {m + 1}. Since S0 is represented but Sm is not, there is some k such that Sk is represented but Sk+1 is not. Let U = Sk ∩Sk+1 , V = nr(Sk ∪Sk+1 ) and V let m ∈ Sk be Vsuch that Sk = U ∪{m} and Sk+1 = U ∪{m +1}. If ø(x) = i∈U ϕ(x, bi )∧ i∈V ¬ϕ(x, bi ), it follows that since Sk is represented, |= ∃x(ø(x) ∧ ϕ(x, bm ) ∧ ¬ϕ(x, bm+1 )) but since Sk+1 is not represented, |= ¬∃x(ø(x) ∧ ϕ(x, bm+1 ) ∧ ¬ϕ(x, bm )). Hence if è(x, y) = ø(x) ∧ ϕ(x, y) we have that è(C, bm+1 ) ( è(C, bm ).

2. ϕ-types, stability and simplicity

17

By indiscernibility, for all rational numbers m ≤ q < q 0 ≤ m + 1, è(C, bq 0 ) ( è(C, bq ) which implies that è(x, y) has the strict order property. a Definition 2.19. Let k ≥ 2 be a natural number. It is said that ϕ(x, y) ∈ L has the k-tree property if there is a tree (as : s ∈ ù <ù ) such that 1. For each f ∈ ù ù , {ϕ(x, afn ) : n < ù} is consistent. 2. For each s ∈ ù <ù the set {ϕ(x, as a i ) : i < ù} is k-inconsistent, that is, every subset of k elements is inconsistent. The formula ϕ has the tree property if it has the k-tree property for some k. Proposition 2.20. If ϕ(x, y) has the strict order property, then ø(x; y1 y2 ) = ¬ϕ(x, y1 ) ∧ ϕ(x, y2 ) has the 2-tree property. Proof. By the strict order property, there is a sequence (ap : p < Q) such that ϕ(C, ap ) ( ϕ(C, aq ) for p < q ∈ Q. We prove the existence of parameters bs = bs1 bs2 , (s ∈ ù <ù ) witnessing the 2-tree property of ø(x; y1 , y2 ). The construction is done by induction on the length of s in such a way that for each s ∈ ù <ù there are ps < qs ∈ Q with aps = bs1 and aqs = bs2 and pt < ps < qs < qt if t ( s. We start with p∅ = 0 and q∅ = 1. To extend the branch finishing in s ∈ ù <ù it is enough to pick two increasing sequences of rational numbers (ps a i : i < ù) and (qs a i : i < ù) such that ps < ps a i < qs a i < ps a i+1 < qs . a Proposition 2.21. Any formula with the tree property is unstable. Proof. Assume ϕ = ϕ(x, y) has the k-tree property. Chose ë such that ëù > 2ù and ëù > ë. By compactness, there is a tree (as : s ∈ ë<ù ) such that for each s ∈ ë<ù , {ϕ(x, as a i ) : i < ë} is k-inconsistent and for each f ∈ ëù , ù ðf (x) = {ϕ(x, afn ) : n < ù} is consistent. S Choose for each f ∈ ë a subset ù If ⊆ ë such that f ∈ If and pf (x) = g∈If ðg (x) is a maximally consistent union of types ðg . By k-inconsistency If is a k-branching tree of height ù and hence |If | ≤ 2ù . Since ëù > 2ù , {pf (x) : f ∈ ëù } has cardinality ëù . Since it is a set of pairwise incompatible ϕ-types over a set of parameters {as : s ∈ ë<ù } of cardinality ë, we conclude that ϕ is not ë-stable. a Definition 2.22. The theory T is ë-stable, or stable in ë, if |Sn (A)| ≤ ë whenever |A| ≤ ë and n < ù. It is enough to check this for n = 1. T is stable if all formulas are stable in T , otherwise it is unstable. Proposition 2.23. The following are equivalent: 1. T is stable. 2. T is ë-stable for all ë such that ë|T | = ë. 3. T is ë-stable for some ë. Q Proof. 1 ⇒ 2. Clear since if x is a n-tuple |Sn (A)| ≤ | ϕ(x,y) Sϕ (A)|. 3 ⇒ 1. If T is stable in ë, then every formula is ë-stable. a

18

2. ϕ-types, stability and simplicity

Definition 2.24. T is simple if it does not have formulas with the tree property. It is said that T has the independence property if some formula has the independence property in T , otherwise we say T has nip or it is dependent. Finally, it is said that T has the strict order property if some formula has the strict order property in T . Remark 2.25. We have seen that 1. T is unstable if and only if T has the independence property or it has the strict order property. 2. Simple theories do not have the strict order property. 3. Stable theories are simple. 4. T is stable if and only if T is simple and does not have the independence property. Remark 2.26. 1. T is stable if and only if T (A) is stable. 2. T is simple if and only if T (A) is simple. Proof. If ϕ(x, y) ∈ L has the order property in T , it also has the order property in T (A). On the other hand, if ϕ(x, y, z) ∈ L, a ∈ A, and ϕ(x, y, a) has the order property in T (A), then ϕ(x; yz) has the order property in T . This proves 1. A similar argument with the k-tree property can be used for 2. a Remark 2.27. 1. T is stable if and only if T eq is stable. 2. T is simple if and only if T eq is simple. Proof. It is evident that if T eq is stable or simple then T is also stable or simple respectively. For the opposite direction use Lemma 1.8. a

Chapter 3

∆-TYPES AND THE LOCAL RANK D(ð, ∆, k)

We generalize the notions of ϕ-formula and ϕ-type to finite sets of formulas ∆. Definition 3.1. Let ∆ = {ϕi (x, yi ) : 1 ≤ i ≤ n} where ϕi (x, yi ) ∈ L for each i. A ∆-formula over A is a formula of the form ϕi (x, a) or ¬ϕi (x, a) with a ∈ A. A ∆-type over A is a consistent set of ∆-formulas over A. A ∆-type p(x) over A is complete if for all i = 1, . . . , n for every a ∈ A, either ϕi (x, a) ∈ p or ¬ϕi (x, a) ∈ p. The set of all complete ∆-types over A is S∆ (A). We endow S∆ (A) with a compact Hausdorff totally disconnected topology. A basis of clopen sets is given by all sets of the form [ø] = {p ∈ S∆ (A) : p ` ø} for any boolean combination ø = ø(x) of ∆-formulas over A. All these notions also apply to the case A = C. Definition 3.2. Let ∆ = {ϕi (x, yi ) : 1 ≤ i ≤ n} and let k < ù. The local D-rank with respect to ∆ and k is defined inductively for any set of formulas ð = ð(x) by the following clauses: 1. D(ð, ∆, k) ≥ 0 if and only if ð is consistent. 2. D(ð, ∆, k) ≥ α + 1 if and only if there is some i (1 ≤ i ≤ n) and there is some sequence (aj : j < ù) such that {ϕi (x, aj ) : j < ù} is k-inconsistent and for all j < ù, D(ð(x) ∪ {ϕi (x, aj )}, ∆, k) ≥ α. 3. D(ð, ∆, k) ≥ α if and only if D(ð, ∆, k) ≥ â for all â < α if α is a limit ordinal. Observe that {α : D(ð, ∆, k) ≥ α} is an initial segment of the ordinals. If ð is inconsistent we set D(ð, ∆, k) = −1; otherwise D(ð, ∆, k) is the supremum of all α such that D(ð, ∆, k) ≥ α. Hence, if D(ð, ∆, k) ≥ α for all α then D(ð, ∆, k) = ∞. In case ∆ = {ϕ(x, y)} we use the notation D(ð, ϕ, k). Remark 3.3. 1. Assume ð(x) ` ð0 (x), ∆ ⊆ ∆0 and k ≤ k 0 . Then D(ð(x), ∆, k) ≤ D(ð0 (x), ∆0 , k 0 ). 2. If ð(x) and ð0 (x) are equivalent, then D(ð(x), ∆, k) = D(ð0 (x), ∆, k). 3. For any set ð(x, y) of formulas over A, for any ∆, k, and α, {a : D(ð(x, a), ∆, k) ≥ α} is type-definable over A. 19

20

3. ∆-types and the local rank D(ð, ∆, k)

Proof. 2 follows from 1, and to prove 1 one shows by induction on α that D(ð(x), ∆, k) ≥ α ⇒ D(ð0 (x), ∆0 , k 0 ) ≥ α. 3 is easily proved by induction on α. a Lemma 3.4. Let ∆ = {ϕ1 (x, y1 ), . . . , ϕn (x, yn )} where ϕi (x, yi ) ∈ L for every i. There is a formula ø∆ = ø∆ (x; y1 , . . . , yn , z, z1 , . . . , z2n ) ∈ L such that 1. For each A with |A| ≥ 2 for each ∆-formula ϕ(x) over A there is a tuple a ∈ A such that ϕ(x) ≡ ø∆ (x; a). 2. For each A for each tuple a ∈ A such that ø∆ (x; a) is consistent, there is a ∆-formula ϕ(x) over A such that ϕ(x) ≡ ø∆ (x; a). Proof. Take as ø∆ (x; y1 , . . . , yn , z, z1 , . . . , z2n ) the following formula: n ^

(z = zi → ϕi (x, yi )) ∧

i=1

n ^

(z = zn+i → ¬ϕi (x, yi )) ∧

i=1 2n _ i=1

(z = zi ) ∧

^

¬(z = zi ∧ z = zj ).

1≤i<j≤2n

Choose a0 , a1 ∈ A such a0 6= a1 . Then for each a ∈ A, ϕi (x, a) is equivalent to ø∆ (x; b1 , . . . , bn , c, c1 , . . . , c2n ) where bi = a for all i = 1, . . . , n, c = a0 = ci and cj = a1 for j 6= i; and ¬ϕi (x, a) is equivalent to ø∆ (x; b1 , . . . , bn , c, c1 , . . . , c2n ) where bi = a for all i = 1, . . . , n, c = a0 = cn+i and cj = a1 for j 6= n + i a Remark 3.5. If ∆ = {ϕ1 (x, y1 ), . . . , ϕn (x, yn )}, each ϕi (x, yi ) is stable, and ø∆ is chosen as in the previous lemma, then ø∆ is stable. Proof. By Corollary 2.12. a Corollary 3.6. For each set ∆ = {ϕ1 (x, y1 ), . . . , ϕn (x, yn )} of formulas ϕi (x, yi ) ∈ L, there is a formula ø∆ (x, z) ∈ L such that for all ð(x), for all k, D(ð, ∆, k) = D(ð, ø∆ , k). Proof. The formula ø∆ is chosen accordingly to Lemma 3.4. By induction on α we see that for each ð and k, D(ð, ∆, k) ≥ α if and only if D(ð, ø∆ , k) ≥ α. This is clear for α = 0 and follows from the induction hypothesis for limit α. The case α + 1 is easy and only requires noticing that ∆ is finite and therefore any infinite sequence of ∆-formulas contains an infinite subsequence of instances of a single formula. a

3. ∆-types and the local rank D(ð, ∆, k)

21

Due to this last result, we will concentrate on the study of D(ð, ϕ, k) rank. The generalization of the statements to D(ð, ∆, k) rank is straightforward. Definition 3.7. Let ϕ(x, y) ∈ L and let k < ù. The formula ϕ(x, a) kdivides over A if there is a sequence (ai : i < ù) such that {ϕ(x, ai ) : i < ù} is k-inconsistent and ai ≡A a for all i < ù. We say that ϕ(x, a) divides over A if it k-divides over A for some k. If A is omitted we understand that A = ∅. Sometimes we say ϕ(x, a) divides over A with respect to k instead of saying that ϕ(x, a) k-divides over A. Proposition 3.8. Let (ϕi (x, yi ) : i < α) be a sequence of L-formulas and let (ki : i < α) be a sequence of natural numbers. For any partial type ð(x) over A, the following are equivalent: 1. There is a sequence (bi : i < α) such that ð(x) ∪ {ϕi (x, bi ) : i < α} is consistent and for each i < α, ϕi (x, bi ) ki -divides over Ab 2|T |+|A|+|α| . By compactness there are as , (s ∈ ë≤α ) such that for each f ∈ ëα , ð(x) ∪ {ϕi (x, afi+1 ) : i < α} is consistent and for each i < α, for each s ∈ ëi , {ϕi (x, as a l ) : l < ë} is ki -inconsistent. Observe that by choice of ë, for any i < α for any s ∈ ëi at least ë of the as a l have the same type over A{asj+1 : j < i}. Hence we can prune the tree obtaining a subtree where this happens for all as a l . Finally choose a branch f ∈ ëα and put bi = afi+1 for all i < α. a Remark 3.9. Assume ð(x) is a partial type over A, ϕ(x, y) ∈ L, and k < ù. If for each n < ù there is a sequence (ai : i < n) such that ð(x)∪ {ϕ(x, ai ) : i < n} is consistent and for each i < n, ϕ(x, ai ) k-divides over Aa
a

22

3. ∆-types and the local rank D(ð, ∆, k)

Lemma 3.10. Let ð(x) be a partial type over A. D(ð(x), ∆, k) ≥ α + 1 if and only if for some ϕ(x, y) ∈ ∆, for some a, D(ð(x) ∪ {ϕ(x, a)}, ∆, k) ≥ α and ϕ(x, a) k-divides over A. Proof. The direction from right to left is obvious from the definitions of D-rank and dividing. For the other direction, assume D(ð(x), ∆, k) ≥ α + 1. Let ë > 2|T |+|A| . From point 3 in Remark 3.3 and compactness, we see that there are ϕ(x, y) ∈ ∆ and (ai : i < ë) such that for each i < ë, D(ð(x) ∪ {ϕ(x, ai )}, ∆, k) ≥ α and {ϕ(x, ai ) : i < ë} is k-inconsistent. By choice of ë, there is an infinite subset I ⊆ ë such that ai ≡A aj for all i, j ∈ I . Then it suffices to take a = ai for some i ∈ I . a Proposition 3.11. For any partial type ð(x) over A, any ϕ = ϕ(x, y) ∈ L, any k < ù, and any ordinal α ≤ ù the following are equivalent: 1. D(ð(x), ϕ, k) ≥ α. 2. There is a sequence (ai : i < α) such that ð(x) ∪ {ϕ(x, ai ) : i < α} is consistent and for each i < α, ϕ(x, ai ) k-divides over Aa
3. ∆-types and the local rank D(ð, ∆, k)

23

l < α} is consistent. Since ϕ(x, al ) also k-divides over Abi a
Chapter 4

FORKING

Definition 4.1. Let ð(x) be a set of formulas over B. We say that ð(x) divides over A if ð implies a formula ϕ(x, a) which divides over A. Notice that we may always assume that a ∈ B and that ϕ(x, a) is a conjunction of formulas in ð(x). Remark 4.2. 1. ϕ(x, a) divides over A if and only if the set {ϕ(x, a)} divides over A. 2. If ð(x) is inconsistent, it divides over A. 3. A partial type ð(x) over acl(A) does not divide over A. 4. ð(x, a) divides over A if and only if for some S A-indiscernible sequence (ai : i < ù) with a0 = a, the set of formulas i<ù ð(x, ai ) is inconsistent. 5. acl(A) = {a : tp(a/Aa) does not divide over A}. 6. ð(x) divides over A if and only if it divides over acl(A). 7. ð(x) divides over A if and only if it divides over some model M ⊇ A. Proof. For 2 take as ϕ(x, y) the formula x 6= x. For 4 use Ramsey’s Theorem to obtain indiscernibility. For 5 consider the formula x = a. For 6 and 7 use Corollary 1.7. a Definition 4.3. The set of formulas ð(x) forks over A if for some n there are formulas ϕ1 (x, a1 ), . . . , ϕn (x, an ) such that ð(x) ` ϕ1 (x, a1 ) ∨ · · · ∨ ϕn (x, an ) and every ϕi (x, ai ) divides over A. The formula ϕ(x, a) forks over A if the set {ϕ(x, a)} forks over A. A nonforking extension of a type p(x) ∈ S(A) over B ⊇ A is a type q(x) ∈ S(B) that extends p and does not fork over A. If an extension q of p forks over A it is called a forking extension of p. Remark 4.4. 1. If ð(x) divides over A, then it forks over A. 2. If ð(x) is finitely satisfiable in A, then it does not fork over A. 3. ð(x) forks over A if and only if a conjunction of formulas in ð(x) forks over A. 4. If a partial type ð(x) over B does not fork over A, then it can be extended to a complete type over B which does not fork over A. 5. ð(x) forks over A if and only if it forks over acl(A). 25

26

4. Forking

6. Let ð(x) be a partial type over A. If ð(x) is algebraic, then no extension of ð(x) forks over A. Proof. The first three points follow directly from the definitions. For 4 check the consistency of ð(x) ∪ {¬ϕ(x) : ϕ(x) ∈ L(B) forks over A} and take as p any complete type over B extending this partial type. For 5 use point 6 of Remark 4.2. 6. Assume ð0 (x) extends ð(x) and forks over A. By 5, ð0 (x) forks over acl(A). But this contradicts 2 since all realizations of ð0 (x) lie in acl(A). a Example 4.5. A formula that forks but does not divide. It is taken from [15], Example 2.11. In a circle consider the ternary relation: B(x, y, z) if and only if y lies between x and z (clockwise) and the arc from x to z is shorter than the arc from z to x. If we choose a0 , a1 , a2 dividing the circle into three equal parts and B(a0 , a1 , a2 ), then the disjunction B(a0 , x, a1 ) ∨ B(a1 , x, a2 ) ∨ B(a2 , x, a0 ) defines the whole circle and hence it does not divide over ∅. But each B(ai , x, aj ) divides over ∅. Therefore x = x forks but does not divide over ∅. Proposition 4.6. Let A ⊆ M and assume for each finite B ⊆ M every finitary type over AB is realized in M . If p(x) ∈ S(M ), then p divides over A if and only if p forks over A. Proof. Assume p(x) forks over A. For some è(x, y) ∈ L, for some finite tuple a ∈ M there are ϕ1 (x, y1 ), . . . , ϕn (x, yn ) ∈ L and finite tuples b1 , . . . , bn such that p(x) ` è(x, a), each ϕi (x, bi ) divides over A, and è(x, a) ` ϕ1 (x, b1 ) ∨ · · · ∨ ϕn (x, bn ). By the assumption on M we may find tuples a1 , . . . , an ∈ M such that a1 , . . . , an ≡Aa b1 , . . . , bn . Then è(x, a) ` ϕ1 (x, a1 ) ∨ · · · ∨ ϕn (x, an ) and each ϕi (x, ai ) divides over A. Since p(x) ` ϕi (x, ai ) for some i, we conclude that p(x) divides over A. a Lemma 4.7. The following are equivalent: 1. tp(a/Ab) does not divide over A. 2. For every infinite A-indiscernible sequence I such that b ∈ I , there is some a 0 ≡Ab a such that I is Aa 0 -indiscernible. 3. For every infinite A-indiscernible sequence I such that b ∈ I , there is some J ≡Ab I such that J is Aa-indiscernible. Proof. The equivalence of 2 and 3 follows by conjugation. It is clear that 3 implies 1. We prove that 1 implies 2. We may assume that A is empty, that I = (bi : i < ù) and that b = b0 . Let p(x, b) = tp(a/b) and let Γ(x, (xi : i < ù)) be the set of all formulas ϕ(x, x0 , . . . , xn ) ↔ ϕ(x, xi0 , . . . , xin )

4. Forking

27

for all ϕ ∈ L and i0 < · · · < in < ù. This set expresses the fact that (xi : i < ù) is x-indiscernible. It will be enough S to prove that p(x, b) ∪ Γ(x, (bi : i < ù)) is consistent. By 1, q(x) = i<ù p(x, bi ) is consistent. Let c |= q and let Γ0 a finite subset of Γ. By Ramsey’s Theorem, there is an order preserving f : ù → ù such that |= Γ0 (c, (bf(i) : i < ù)). By indiscernibility (bi : i < ù) ≡ (bf(i) : i < ù) and therefore we can find c 0 such that c 0 (bi : i < ù) ≡ c(bf(i) : i < ù). Clearly c 0 |= q(x) ∪ Γ0 (x, (bi : i < ù)). a Proposition 4.8. If tp(a/B) does not divide over A ⊆ B and tp(b/Ba) does not divide over Aa, then tp(ab/B) does not divide over A. Proof. It is an easy application of point 3 of Lemma 4.7. a Proposition 4.9. If ϕ(x, a) k-divides over A and tp(b/Aa) does not divide over A, then ϕ(x, a) k-divides over Ab. Proof. Let I = (ai : i < ù) be an A-indiscernible sequence such that a = a0 and {ϕ(x, ai ) : i < ù} is k-inconsistent. By Lemma 4.7 there is some J ≡Aa I which is Ab-indiscernible. Then J witnesses that ϕ(x, a) divides over Ab with respect to k. a Definition 4.10. A dividing chain for ϕ(x, y) is a sequence (ai : i < α) such that {ϕ(x, ai ) : i < α} is consistent and for every i < α, ϕ(x, ai ) divides over a
28

4. Forking

3. There is no increasing chain (pi (x) : i < |T |+ ) of types pi (x) ∈ S(Ai ) such that for every i < |T |+ , pi+1 divides over Ai . 4. For some cardinal κ ≥ |T | there is no increasing chain (pi (x) : i < κ + ) of types pi (x) ∈ S(Ai ) such that for every i < κ + , pi+1 divides over Ai . Proof. Simplicity implies 1, since if p ∈ S(A) divides over every subset of A of cardinality ≤ |T |, then we can inductively construct a sequence of formulas (ϕi (x, yi ) : i < |T |+ ) and a sequence (ai : i < |T |+ ) of parameters ai ∈ A such that ϕi (x, ai ) ∈ p and ϕi (x, ai ) divides over a
4. Forking

29

D(pi (x), ∆, k) > D(pi+1 (x), ∆, k) for all i < |T |+ , which is a contradiction since the local D-rank is finite in a simple theory. a Corollary 4.16. If T is simple and p(x) ∈ S(A), then p does not fork over A. Hence, for any B ⊇ A there is a nonforking extension q(x) ∈ S(B) of p. Proof. It is enough to check the first assertion for finitary types and this case follows from Proposition 4.15. The rest is point 4 in Remark 4.4. a

Chapter 5

INDEPENDENCE

Definition 5.1. We say that A is independent of B over C (written A ^ | C B) if for every finite tuple a ∈ A, tp(a/BC ) does not fork over C . A ^ 6 | C B means that not A ^ | C B. In the case C = ∅ we write A ^ | B and A ^ 6 | B. Remark 5.2. If instead of sets A, B, C we put partially, or everywhere, tuples or sequences a, b, c in the independence relation we mean the independence of the enumerated sets. But it is easy to prove that A ^ | C B if and only if tp(a/BC ) does not fork over C for some (every) enumeration a of A. Remark 5.3. The independence relation always has the following properties: 1. Invariance: If f ∈ Aut(C) and A ^ | C B, then f(A) ^ | f(C ) f(B). 2. Normality: A ^ | C B if and only if A ^ | C CB if and only if AC ^ | C B. 3. Finite character: If a ^ | C b for all finite tuples a ∈ A, b ∈ B, then A^ | C B. 4. Base monotonicity: If A ^ | C B and B 0 ⊆ B, then A ^ | CB 0 B. 5. Monotonicity: If A ^ | C B, A0 ⊆ A and B 0 ⊆ B, then A0 ^ | C B 0. 6. Anti-reflexivity: If A ^ | B A, then A ⊆ acl(B). 7. Algebraic closure: acl(A) ^ | A B. Proof. Only the implication A ^ | C B ⇒ AC ^ | C B needs some checking. For this note that if ϕ(x, y, z) ∈ L, c ∈ C , and ϕ(x, y, d ) divides over C then also ϕ(x, c, d ) divides over C . For point 6 use point 5 of Remark 4.2. Finally, for point 7 use point 6 of Remark 4.4. a Proposition 5.4. Let A ⊆ M and assume for every finite B ⊆ M , every finitary type over AB is realized in M . If a ^ | A M and b ^ | Aa M , then ab ^ | A M. a

Proof. By Lemma 4.8 and Proposition 4.6.

Proposition 5.5 (Local character). Let T be simple. For any B, C there is some A ⊆ B such that |A| ≤ |T | + |C | and C ^ | A B. Proof. This is clear for finite C by Proposition 4.15. For the general case, choose first some AX ⊆ B such that |AX | ≤ |T | and X ^ | A B for each finite X

31

32

5. Independence

X ⊆ C and let A be the union of all these sets AX . Then |A| ≤ |T | + |C | and C^ | A B. a Proposition 5.6 (Closedness). The set of all complete types p(x) ∈ S(B) which do not fork over A is a closed set in S(B). Proof. Let ð(x) be the set of all negations ¬ϕ(x) of all formulas ϕ(x) ∈ L(B) which fork over A. Then p(x) ∈ S(B) does not fork over A if and only if p extends ð. a Proposition 5.7 (Extension). Let T be simple and let a be a possibly infinite tuple. For any B, there is some a 0 ≡A a such that a 0 ^ | A B. Proof. By Corollary 4.16. The infinite case is also covered. a Remark 5.8. A type p(x) ∈ S(B) which does not fork over A ⊆ B has also a global nonforking extension p(x) ∈ S(C) which does not fork over A. Therefore, in a simple theory any type has a global nonforking extension. Proof. The same argument as for a nonforking extension over a small set. a Definition 5.9. Let (I, <) be a linearly ordered set. The sequence (ai : i ∈ I ) is A-independent (or independent over A) if for every i ∈ I , ai ^ | a
A Morley sequence over A is a sequence (ai : i ∈ I ) which is A-independent and A-indiscernible. It is said to be a Morley sequence in the type p ∈ S(A) if it is a Morley sequence over A and every ai realizes p. Remark 5.10. Let (I, <) be an infinite linearly ordered set and let (ai : i ∈ I ) be an A-indiscernible sequence. The sequence is nontrivial (i.e., ai 6= aj for all i 6= j) if and only if tp(ai /A) is nonalgebraic. Lemma 5.11. If p(x) ∈ S(B) does not fork over A ⊆ B, there is a Morley sequence (ai : i < ù) in p which is moreover a Morley sequence over A. Proof. Let α be the length of x, let κ = |B| + |T | + |α| and ë = i(2κ )+ . Since p(x) does not fork over A, one can construct a sequence (ai : i < ë) of realizations ai of p such that ai ^ | A Ba
5. Independence

33

Remark 5.12. Let p(x) ∈ S(A). If there is a Morley sequence (ai : i < ù) in p, then for any linearly ordered set (I, <) there is a Morley sequence (bi : i ∈ I ) in p. It is enough to obtain (bi : i ∈ I ) as an A-indiscernible sequence with the same Ehrenfeucht–Mostowski set over A (the set of formulas over A realized by increasing finite tuples) as (ai : i < ù). Proposition 5.13. If a = (ai : i < ù) is a Morley sequence over A and B ⊇ A, there is a Morley sequence b = (bi : i < ù) over B such that a ≡A b. Proof. Let α be the length of each ai , let κ = |B|+|T |+|α| and ë = i(2κ )+ . Extend a to an A-indiscernible sequence (ai : i < ë). It is also a Morley sequence over A. Construct inductively a sequence (ai0 : i < ë) such that 0 0 for all i < ë, a
Lemma 5.14. Let (ai : i ∈ I ) be A-independent. If J , K are subsets of I such that J < K (that is, j < k for any j ∈ J , k ∈ K ), then tp((ai : i ∈ K )/A(ai : i ∈ J )) does not divide over A. Proof. It can be assumed that K is finite. An induction on |K | using Lemma 4.8 easily gives the result. a Proposition 5.15. Let T be simple, and let ð(x, y) be a set of formulas over A. Then ð(x, a) divides over S A if and only if for every Morley sequence (ai : i < ù) over A in tp(a/A), i<ù ð(x, ai ) is inconsistent. Proof. Without loss of generality A = ∅ and ð(x, y) = {ϕ(x, y)}. Assume that ϕ(x, a) divides over ∅ but for some infinite Morley sequence the inconsistency fails. Let (I, <) be a linearly ordered set isomorphic to the reverse order of the cardinal |T |+ . By compactness there is a Morley sequence aI = (ai : i ∈ I ) in tp(a) such that {ϕ(x, ai ) : i ∈ I } is consistent. Let c realize this type. By simplicity there is J ⊆ I of cardinality at most |T | such that tp(c/aI ) does not fork over aJ = (ai : i ∈ J ). By choice of the order of I we can find i ∈ I such that i < J . By Lemma 5.14 tp(aJ /ai ) does not divide over ∅. Since ϕ(x, ai ) divides over ∅, by Proposition 4.9 it divides over aJ . But tp(c/aI ) contains ϕ(x, ai ) and hence it divides (and forks) over aJ , a contradiction. a

34

5. Independence

Remark 5.16. The previous result can be easily generalized to Morley sequences (ai : i ∈ I ) for any infinite linear ordering (I, <). Proof. By compactness.

a

Proposition 5.17. Let T be simple. A partial type ð(x) divides over A if and only if it forks over A. Proof. We may assume ð(x) is a formula ϕ(x, a). Assume ϕ(x, a) does not divide over A but it implies a disjunction ϕ1 (x, a1 ) ∨ · · · ∨ ϕn (x, an ) where every ϕi (x, ai ) divides over A. Let (a j a1j . . . anj : j < ù) be a Morley sequence in tp(aa1 . . . an /A). Then (a j : j < ù) is an A-indiscernible sequence of realizations of tp(a/A). By definition of dividing, there exists a realization c of {ϕ(x, a j ) : j < ù}. For every j < ù there is some i such that c realizes some ϕi (x, aij ). By the pigeonhole principle, there is some i such that for an infinite subset J ⊆ ù, c realizes every ϕi (x, aij ) with j ∈ J . By indiscernibility, {ϕi (x, aij ) : j < ù} is consistent and then by Proposition 5.15 ϕi (x, ai ) does not divide over A since (aij : j < ù) is a Morley sequence in tp(ai /A). a Proposition 5.18 (Symmetry). In a simple theory independence is a symmetric relation, i.e, A ^ | C B implies B ^ | C A. Proof. By Proposition 5.17, it is enough to prove that if tp(a/Cb) does not fork over C , then tp(b/Ca) does not divide over C . We may assume that tp(a/C ) is not algebraic. By Lemma 5.11 there is a Morley sequence I = (ai : i < ù) in tp(a/C ) which is Cb-indiscernible and starts with a0 = a. Let ϕ(x, y, z) ∈ L be a formula and c ∈ C such that |= ϕ(a, b, c). We will show that ϕ(a, y, c) does not divide over C . By indiscernibility of I over Cb we know that |= ϕ(ai , b, c) for all i < ù. Hence {ϕ(ai , y, c) : i < ù} is consistent. Since (ai c : i < ù) is a Morley sequence in tp(ac/C ), by Proposition 5.15 we conclude that ϕ(a, y, c) does not divide over C . a Proposition 5.19 (Transitivity). In a simple theory independence is a transitive relation, i.e, whenever B ⊆ C ⊆ D, A ^ | B C and A ^ | C D, then A ^ | B D. Proof. It is a direct consequence of propositions 5.18, 4.8, and 5.17.

a

Proposition 5.20. If T is simple, the independence relation has the following additional properties: 1. 2. 3. 4.

Pairs Lemma: ab ^ | A B if and only if a ^ | A B and b ^ | Aa B. Change of base: if ab ^ | A B, then a ^ | Ab ⇔ a^ | AB b. A^ | B acl(B). A^ | B C ⇔ acl(A) ^ | B C ⇔ A^ | B acl(C ) ⇔ A ^ | acl(B) C .

Proof. It is an easy exercise.

a

Corollary 5.21. Let T be simple. If I is a linearly ordered set and (ai : i ∈ I ) is an A-independent sequence, then ai ^ | A {aj : j 6= i} for all i ∈ I .

5. Independence

35

Proof. By induction on n it is easy to show that ain+1 ^ | A ai1 , . . . , ain for all different i1 , . . . , in+1 ∈ I . For the induction one uses symmetry and Lemma 4.8. a Proposition 5.22. Let T be simple, p(x) ∈ S(A) and let ð(x) be a partial type over B. Then p(x) ∪ ð(x) does not fork over A if and only if D(p, ∆, k) = D(p ∪ ð, ∆, k) for all finite ∆, k. Proof. We may assume A ⊆ B and ∆ = {ϕ}. Direction from right to left follows from Lemma 4.14. Now assume p ∪ ð is a nonforking extension of p and choose q(x) ∈ S(B) a type which does not fork over A and extends p ∪ ð. We will check that D(q, ϕ, k) ≥ D(p, ϕ, k) for all ϕ, k. From this it will follow that D(p, ϕ, k) ≤ D(p ∪ ð, ϕ, k) for all ϕ, k. We freely use transitivity and symmetry of independence and also the fact that dividing and forking coincide. Let n = D(p, ϕ, k). There is a sequence b
Chapter 6

THE LOCAL RANK CB∆ (ð)

Definition 6.1. Let ð(x) be a set of formulas over the set A and let ∆ = {ϕ1 (x, y1 ), . . . , ϕn (x, yn )} be a finite set of formulas ϕi (x, yi ) ∈ L. Let m < ù be the length of the tuple of variables x. Since the restriction map Sm (C) → S∆ (C) is closed and the class Xð,∆ = {p ∈ S∆ (C) : p(x) ∪ ð(x) is consistent} is the image of the closed class {p ∈ Sm (C) : ð(x) ⊆ p}, Xð,∆ is closed in S∆ (C). We define the ∆-rank CB∆ (ð) as the Cantor–Bendixson rank of Xð,∆ in S∆ (C) and the ∆-multiplicity Mlt∆ (ð) as its Cantor–Bendixson degree. Lemma 6.2. If ð1 (x) ` ð2 (x), then CB∆ (ð1 ) ≤ CB∆ (ð2 ) and in case CB∆ (ð1 ) = CB∆ (ð2 ), then Mlt∆ (ð1 ) ≤ Mlt∆ (ð2 ). Proof. Clear, because if Xði ,∆ = {p ∈ S∆ (C) : p is consistent with ði }, then Xð1 ,∆ ⊆ Xð2 ,∆ . a Remark 6.3. For each ∆ = {ϕ1 (x, y1 ), . . . , ϕn (x, yn )} where ϕi (x, yi ) ∈ L for every i, there is a formula ø∆ (x; z) ∈ L such that for each partial type ð(x), CB∆ (ð) = CBø∆ (ð) and Mlt∆ (ð) = Mltø∆ (ð). Proof. By Lemma 3.4. a Proposition 6.4. Let ø(x) be a boolean combination of ∆-formulas. 1. CB∆ (ø) ≥ 0 if and only if ø is consistent. 2. CB∆ (ø) ≥ α + 1 if and only if there is a sequence (øi (x) : i < ù) of pairwise contradictory boolean combinations øi (x) of ∆-formulas such that CB∆ (ø(x) ∧ øi (x)) ≥ α for all i < ù. 3. CB∆ (ø) ≥ α for limit α if and only if CB∆ (ø) ≥ â for all â < α. 4. If CB∆ (ø) = α < ∞, then Mlt∆ (ø) is the largest n < ù for which there is a sequence (øi (x) : i < n) of pairwise contradictory boolean combinations øi (x) of ∆-formulas such that CB∆ (ø(x) ∧ øi (x)) ≥ α for all i < n. The formulas øi in 2 and 4 can be chosen as explicitly contradictory conjunctions of ∆-formulas. Moreover in 2 we can fix some ϕ ∈ ∆ such that each øi is a conjunction of ϕ-formulas. 37

38

6. The local rank CB∆ (ð)

Proof. Let X∆,ø be the clopen subset of S∆ (C) consisting of all types p ∈ S∆ (C) such that p ` ø. CB∆ (ø) is the maximal Cantor–Bendixson rank (in S∆ (C)) of the points in X∆,ø . Points 1 and 3 are clear. The proof of 4 is similar to the proof of 2, so we restrict ourselves to 2. Assume first there is a sequence (øi (x) : i < ù) of pairwise contradictory boolean combinations øi of ∆-formulas such that CB∆ (ø ∧ øi ) ≥ α for each i < ù. For each i choose some pi ∈ S∆ (C) of Cantor–Bendixson rank ≥ α such that pi ` ø ∧ øi . Since the øi are pairwise contradictory, all the pi are different. Since X∆,ø contains infinitely many points of rank ≥ α, it contains some point of rank ≥ α + 1. Hence CB∆ (ø) ≥ α + 1. For the other direction, assume CB∆ (ø) ≥ α + 1. Then X∆,ø is an open set containing a point of rank ≥ α + 1. Thus the set Y0 of points of X∆,ø of rank ≥ α is infinite. Clearly, for some ∆-formula è there are points in Y0 containing è and points in Y0 containing ¬è and one of them, say the second one, is infinite. Let è0 = è and let Y1 be the infinite subset of Y0 consisting of all points containing ¬è0 . Now assume that Yi , øi are defined for all i ≤ n, assume that the Yi build a strictly descending chain of infinite sets, and assume that Yi+1 is the subset of Yi consisting of all its points containing the ∆-formula ¬èi . Again, there is some ∆-formula èn+1 such that some points of Yn+1 contain èn+1 and infinitely many points of Yn+1 contain ¬èn+1 . For some infinite subset I ⊆ ù there is a ϕ ∈ ∆ such that for each i ∈ I , èi isVa ϕ-formula. Without loss of generality, I = ù. We then put øn = èn ∧ i
6. The local rank CB∆ (ð)

39

Proof. Clearly 2 implies 3. Stability of every ϕ ∈ ∆ means that for each infinite set A, |S∆ (A)| ≤ |A|. It is therefore equivalent to the stability of the formula ø∆ given by Lemma 3.4. Hence we may assume that ∆ = {ϕ}. By Proposition 2.6, the stability of ϕ is equivalent to the inconsistency of the set of formulas Γϕ (ù), where for each ordinal α, Γϕ (α) = {ϕ(xf , yfy )f(i) : f ∈ 2α , i < α} and where ϕ 0 = ϕ and ϕ 1 = ¬ϕ. 1 ⇒ 2. Assume CBϕ (x = x) ≥ ù. If ø(x) is a boolean combination of ϕ-formulas and CBϕ (ø) ≥ n + 1 then for some a, CBϕ (ø(x) ∧ ϕ(x, a)) ≥ n and CBϕ (ø ∧ ¬ϕ(x, a)) ≥ n. Since CBϕ (x = x) ≥ ù this can be used to construct a binary tree of parameters (as : s ∈ 2
and choose s for which øs has minimal CBϕ -rank and least multiplicity Mltϕ among the formulas with the same rank. But øs (x) is equivalent to (øs a 0 (x) ∨ øs a 1 (x)) and the formulas øs a 0 , øs a 1 are incompatible. So one of them has smaller CBϕ -rank or they have the same rank and one has smaller ϕ-multiplicity, a contradiction. a Remark 6.7. Let ϕ = ϕ(x, y) ∈ L be stable, let ð(x) be a partial type over A, and let p ∈ Sϕ (C) be consistent with ð(x) and of Cantor–Bendixson rank CBϕ (ð). Then p is definable over acleq (A). If Mltϕ (ð) = 1 it is also A-definable. Proof. By stability of ϕ, p is definable (see Remark 2.7). All the Aconjugates of p have Cantor–Bendixson rank CBϕ (ð) and are consistent with ð(x); their number is bounded by Mltϕ (ð) < ù. Since p has finitely many A-conjugates, by Proposition 1.11 p is acleq (A)-definable. If Mltϕ (ð) = 1, p is A-invariant and therefore A-definable. a Lemma 6.8. Let ϕ(x, y) ∈ L be stable. Let p(x) ∈ Sϕ (C) and q(y) ∈ Sϕ −1 (C) and let dp xϕ(x, y) and dq yϕ(x, y) be corresponding definitions of p and q which are boolean combinations of ϕ −1 -formulas and of ϕ-formulas respectively. Then q ` dp xϕ(x, y) if and only if p ` dq yϕ(x, y). Proof. Let A be a set containing all the parameters of the formulas dp xϕ(x, y) and dq yϕ(x, y) defining respectively p and q. Let (an : n ∈ ù) and (bn : n ∈ ù) be sequences such that an |= p  A{bi : i < n} and bn |= q  A{ai : i ≤ n}. If q ` dp xϕ(x, y) and p 6` dq yϕ(x, y), we would have |= ϕ(am , bn ) if and only if m > n, and therefore ϕ(x, y) would have the order property. a

40

6. The local rank CB∆ (ð)

Definition 6.9. A generalized ϕ-formula over A is a formula over A which is equivalent to a boolean combination of ϕ-formulas, possibly with parameters not in A. A generalized ϕ-type over A is a consistent set of generalized ϕformulas over A. A generalized ϕ-type p(x) over A is complete if for every generalized ϕ-formula ø(x) over A, ø(x) ∈ p or ¬ø(x) ∈ p. The set of all complete generalized ϕ-types over A is Sϕ∗ (A). The generalized ϕ-type of a over A is tp∗ϕ (a/A), the complete generalized ϕ-type which consists of all generalized ϕ-formulas over A realized by a. As in the cases of Sϕ (A) and S(A), we can treat Sϕ∗ (A) as a boolean topological space with basis of clopen sets given by [ø(x)] = {p(x) ∈ Sϕ∗ (A) : ø ∈ p} for every generalized ϕ-formula ø(x) over A. The restriction mappings S(A) → Sϕ∗ (A) and Sϕ∗ (A) → Sϕ (A) are surjective and continuous. Remark 6.10. For any model M , every generalized ϕ-formula over M is equivalent to a boolean combination of ϕ-formulas over M . Hence, every complete ϕ-type over M can be uniquely extended to a complete generalized ϕtype over M and the restriction map Sϕ∗ (M ) → Sϕ (M ) is an homeomorphism. Proof. Assume ø(x) ∈ L(M ) is equivalent to a boolean combination è(x, c1 , . . . , cn ) of ϕ-formulas and è(x, x1 , . . . , xn ) ∈ L. Then M |= ∃x1 . . . xn ∀x(ø(x) ↔ è(x, x1 , . . . , xn )) and therefore we can find c10 , . . . , cn0 ∈ M such that ø(x) ≡ è(x, c10 , . . . , cn0 ). a Remark 6.11. If ϕ(x, y) is stable and p ∈ Sϕ (C) is definable over A, then it is definable by a generalized ϕ −1 -formula over A. Proof. By Remark 2.13. a ∗ ∗ Lemma 6.12. Assume p(x) ∈ Sϕ (A) and q(x) ∈ Sϕ (acl(A)) is an extension of p. Then q(x) is consistent with any extension p 0 (x) ∈ S(A) of p. Proof. Suppose p 0 (x) ∈ S(A) extends p(x) and is inconsistent with q(x). Then for some ø(x) ∈ q(x), p 0 (x) ` ¬ø(x). If ø 0 (x) is an A-conjugate of ø(x), then also p0 (x) ` ¬ø 0 (x). Since ø(x) ∈ L(acl(A)), it has only finitely many A-conjugates and we can form their disjunction è(x). Since è(x) is (equivalent to) a generalized ϕ-formula over A and p 0 (x) ` ¬è(x) we conclude ¬ø(x) ∈ q, a contradiction. a Proposition 6.13. Let ϕ be stable. 1. If p(x) ∈ Sϕ (M ), then there is a unique p(x) ∈ Sϕ (C) extending p which is definable over M and hence Mltϕ (p) = 1. 2. If A = acleq (A) and p(x) ∈ Sϕ∗ (A), there is a unique p(x) ∈ Sϕ (C) consistent with p and definable over A and hence Mltϕ (p) = 1. Proof. 1 follows from Lemma 2.4, but also can be considered a particular case of 2.

6. The local rank CB∆ (ð)

41

2. Existence follows from Remark 6.7. For uniqueness, let p1 , p2 ∈ Sϕ (C) be global ϕ-types consistent with p and A-definable. By Remark 6.11 there are corresponding definitions øi (y) ∈ L(A) (i = 1, 2) which are generalized ϕ −1 formulas over A. Recall that ϕ −1 is also stable. Let b be a tuple of the same length as y and let us choose (by Remark 6.7) a global type q(y) ∈ Sϕ −1 (C) consistent with tp(b/A) and definable over A by a formula è(x) ∈ L(A). By Remark 6.11 again, we may assume è(x) is a generalized ϕ-formula over A. We apply now Lemma 6.8 with øi0 (y) = dpi xϕ(x, y) and è(x) = dq yϕ(x, y) obtaining: ϕ(x, b) ∈ pi ⇔ |= øi (b)

because øi defines pi

⇔ øi (y) ∈ tp(b/A) because øi (y) ∈ L(A) ⇔ q ` øi (y) because q(y) ∪ tp(b/A) is consistent and øi is equivalent to a boolean combination of ϕ −1 -formulas ⇔ pi ` è(x) by Lemma 6.8 ⇔è ∈p

since p(x) ∪ pi is consistent and è is a generalized ϕ-formula over A.

This shows that p1 = p2 . a ∗ Corollary 6.14. Let ϕ = ϕ(x, y) ∈ L be stable and let p(x) ∈ Sϕ (A). Every two types p(x), q(x) ∈ Sϕ (C) consistent with p(x) and definable over acleq (A) are A-conjugate. Proof. Let p, q be two such types. They can be considered complete generalized ϕ-types over C. Let p1 , q1 ∈ Sϕ∗ (acleq (A)) be their corresponding restrictions. By Lemma 6.12 (applied in T eq ) there is some p0 (x) ∈ S(A) extending p which is consistent with p1 (x) and is consistent with q1 (x). Now let p10 , q10 ∈ S(acleq (A)) be extensions of p0 ∪ p1 and of p0 ∪ p2 respectively. Clearly there is some f ∈ Aut(C/A) sending p10 to q10 . Then q and pf are acleq (A)-definable and consistent with q1 . By Proposition 6.13 pf = q. a ∗ Corollary 6.15. Let ϕ = ϕ(x, y) ∈ L be stable, let p(x) ∈ Sϕ (A). For every p(x) ∈ Sϕ (C) consistent with p, the following are equivalent: 1. p is definable over acleq (A). 2. p is a point of Cantor–Bendixson rank CBϕ (p). In case Mltϕ (p) = 1 there is a unique such p ∈ Sϕ and it is in fact A-definable. Proof. Let Xp,ϕ ⊆ Sϕ (C) be the class of all global ϕ-types consistent with p. By Remark 6.7 we know that all types in Xp,ϕ of rank CBϕ (p) are definable over acleq (A). Now let p, q ∈ Xp,ϕ be such that p is acleq (A)-definable and q has Cantor–Bendixson rank CBϕ (p). By Corollary 6.14 they are A-conjugate and therefore p has also rank CBϕ (p) in Xp,ϕ . a

Chapter 7

HEIRS AND COHEIRS

Definition 7.1. Let M ⊆ A and p(x) ∈ S(A). We say that p is an heir of p  M or that p inherits from M if for every ϕ(x, y) ∈ L(M ) if ϕ(x, a) ∈ p for some tuple a ∈ A, then ϕ(x, m) ∈ p for some tuple m ∈ M . We say that p is a coheir of p  M or that p coinherits from M if p is finitely satisfiable in M . The same definitions apply to global types, i.e, to the case A = C. These definitions also make sense for types in infinitely many variables. Remark 7.2. tp(a/Mb) inherits from M if and only if tp(b/Ma) coinherits from M . Proof. It is just a matter of writing down the definitions. a Lemma 7.3. 1. If p(x) ∈ S(M ), then p inherits and coinherits from M . 2. If M ⊆ A and p(x) ∈ S(A) coinherits from M , then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q coinherits from M . 3. If M ⊆ A and p(x) ∈ S(A) inherits from M , then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q inherits from M . Proof. 1 is clear. For 2 it is enough to check the consistency of the following set of formulas p(x) ∪ {¬ϕ(x) : ϕ(x) ∈ L(B) is not satisfiable in M }. 3. In this case it suffices to prove that the following set of formulas is consistent p(x) ∪ {¬ϕ(x, b) : ϕ(x, y) ∈ L(M ), b ∈ B and there is no m ∈ M such that ϕ(x, m) ∈ p  M }.

a

Definition 7.4. Let p(x) ∈ S(B) and let A ⊆ B. We say that p splits over A if for some ϕ(x, y) ∈ L(A) there are a, b ∈ B such that a ≡A b, ϕ(x, a) ∈ p and ¬ϕ(x, b) ∈ p. This applies also to the case B = C. Note that the same notion is defined if one requires ϕ(x, y) ∈ L. If p ∈ S(C), then clearly p does not split over A if and only if pf = p for each f ∈ Aut(C/A). If moreover A = M , a global nonsplitting extension is also sometimes called an M -special extension. 43

44

7. Heirs and coheirs

Proposition 7.5. 1. The number of global nonsplitting extensions of a |A|+|T | finitary type p ∈ S(A) is ≤ 22 . 2. Let p be a global nonsplitting extension of p ∈ S(A). If the sequence (ai : i < α) is constructed in such a way that for all i < α, ai |= p  Aa
7. Heirs and coheirs

45

5. Let T be simple and assume p(x) ∈ S(A) inherits from M ⊆ A. Let a ∈ A be a tuple and let b |= p. We want to show that b ^ | M a. By Remark 7.2 tp(a/Mb) coinherits from M and by point 3, a ^ | M b. The result then follows by symmetry of independence. a Definition 7.7. Given p(x) ∈ S(M ), by (M, dp) we refer to the expansion of M to language L ∪ {Rϕ : ϕ ∈ L} where for every ϕ = ϕ(x, y) ∈ L, if y = y1 , . . . , yn then Rϕ is n-ary and it is interpreted as {a ∈ M : ϕ(x, a) ∈ p}. Let M  N and p(x) ⊆ q(x) ∈ S(N ). We say that q is a strong heir of p if (M, dp)  (N, dq). This also makes sense when N = C. Remark 7.8. 1. Strong heirs are heirs. 2. If (M, dp)  N 0 and N = N 0  L, then for some q(x) ∈ S(N ), p ⊆ q and N 0 = (N, dq). 3. Any strong heir of a nondefinable type is again nondefinable. Proof. 1 and 2 are easy. For 3, let q(x) ∈ S(N ) be a strong heir of p, and assume ϕ(x, y) ∈ L, ø(y, z) ∈ L, n ∈ N and ø(y, n) defines q  ϕ. Then (N, dq) |= ∃z∀y(ø(y, z) ↔ Rϕ (y)). Since (M, dp)  (N, dq), for some m ∈ M , (M, dp) |= ∀y(ø(y, m) ↔ Rϕ (y)). Then ø(y, m) defines p  ϕ. a Proposition 7.9. If p(x) ∈ S(M ) is not definable, then p(x) has unboundedly many (nondefinable) strong heirs over C. Proof. We first show that p(x) has two strong heirs over some N  M . Since p is not definable, (M, dp) is not a definable expansion of M . By Svenonius’s Theorem (see, for instance, Th´eor`eme 9.02 in [36]), there is some N 0  (M, dp) having some f ∈ Aut(N 0  L/M ) such that f 6∈ Aut(N 0 ). Let N = N 0  L. Then for some q(x) ∈ S(N ), N 0 = (N, dq) and q f 6= q. Clearly q and q f are two strong heirs of p. Since a strong heir of a nondefinable type is again nondefinable, we can iterate this procedure (taking unions at limits) obtaining for each cardinal κ a family (pi (x) : i < κ) of strong heirs pi ∈ S(Mi ) of p such that pi ∪ pj is inconsistent if i 6= j. Clearly, each pi can be extended to a type pi over C which is a strong heir of pi and therefore also of p. a Definition 7.10. A coheir sequence over A is a sequence (ai : i < α) such that for some M ⊆ A, for all i < j < α, tp(ai /Aa
46

7. Heirs and coheirs

2. Choose an extension p ∈ S(C) of p which coinherits from M and choose ai |= p  Aa
Chapter 8

STABLE FORKING

Proposition 8.1. Let ∆ = {ϕi (x, yi ) : i < n} be a set of stable formulas. A type p ∈ S∆ (C) is definable over a model M if and only if it is finitely satisfiable in M . In fact, if p is M -definable and it is consistent with a partial type ð(x) over M , then ð(x) ∪ p(x) is finitely satisfiable in M . Proof. We may assume ∆ = {ϕ(x, y)}. Let p be M -definable and let us choose by Remark 2.13 a definition dp xϕ(x, y), which is a boolean combination of ϕ −1 -formulas over M . Let ϕ(x, a1 ), . . . , ϕ(x, an ), ¬ϕ(x, b1 ), . . . , ¬ϕ(x, bm ) be formulas in p. For 1 ≤ i ≤ n and 1 ≤ j ≤ m, let qi = tpϕ −1 (ai /M ) and rj = tpϕ −1 (bj /M ). Again by Remark 2.13 there are qi ∈ Sϕ −1 (C) and rj ∈ Sϕ −1 (C) extending qi and rj respectively and having definitions dqi yϕ(x, y) and drj yϕ(x, y) which are boolean combinations of ϕ-formulas over M . Then |= dp xϕ(x, ai ) and |= ¬dp xϕ(x, bj ) and hence qi ` dp xϕ(x, y) and rj ` ¬dp xϕ(x, y). By Lemma 6.8, p ` dqi yϕ(x, y) and p ` ¬drj yϕ(x, y). Since these are formulas over M , for some c ∈ M , |= dqi yϕ(c, y) and |= ¬drj yϕ(c, y) for all i, j. Then |= ϕ(c, ai ) and |= ¬ϕ(c, bj ) for all i, j. Clearly this c can also be found realizing additionally a given finite subset of ð(x). For the other direction, let us assume dxϕ(x, y) is a definition of p which is not equivalent to a formula over M . Then we can find b, c such that b ≡M c and |= dxϕ(x, b) but |= ¬dxϕ(x, c). In this case ϕ(x, b) ∈ p and ¬ϕ(x, c) ∈ p but there is no a ∈ M such that |= ϕ(a, b) ∧ ¬ϕ(a, c). Hence p is not finitely satisfiable in M . a Proposition 8.2. Let ϕ(x, y) ∈ L be stable, let p(x) ∈ Sϕ (C) and assume p is definable over M and it is consistent with ð(x), a partial type over M . For some q(x) ∈ S(M ) extending ð(x) ∪ p  M there is a Morley sequence (ci : i < ù) in q such that p is definable by a positive boolean combination of the formulas ϕ(ci , y). Proof. By Proposition 8.1 ð(x) ∪ p(x) is finitely satisfiable in M . It is easy to check the consistency of ð(x) ∪ p(x) ∪ {¬ø(x) : ø(x) ∈ L(C) is not satisfiable in M }. 47

48

8. Stable forking

Let q ∈ S(C) be an extension of this set of formulas. Clearly q coinherits from M and q  ϕ = p. We claim that for some n < ù there is a sequence (ci : i < n) such that ci |= q  Mc
8. Stable forking

49

4 ⇒ 5. The same reason, since in fact every ϕ(x, ci ) is an acleq (A)-conjugate of ϕ(x, a). 5 ⇒ 6. Clear by construction of ó. 6 ⇒ 7. ó(x) satisfies the requirements in 7. 7 ⇒ 1. Let ó 0 (x) be a positive boolean combination of A-conjugates of ϕ(x, a) which is equivalent to a formula over A and is consistent with ð. By Remark 6.7 there is p ∈ Sϕ (C) definable over acleq (A) and consistent with ð(x) ∪ {ó 0 (x)}. Since ó 0 (x) is a disjunction of conjunctions of A-conjugates of ϕ(x, a), some A-conjugate of ϕ(x, a) appears in p. By conjugation over A, there is also some p0 ∈ Sϕ (C) definable over acleq (A) and consistent with ð(x) such that ϕ(x, a) ∈ p0 . a Corollary 8.4. For any stable formula ϕ(x, y) ∈ L, the following are equivalent. 1. 2. 3. 4. 5.

ϕ(x, a) does not divide over A. ϕ(x, a) is satisfiable in every model M ⊇ A. Some p ∈ Sϕ (C) containing ϕ(x, a) is definable over acleq (A). ϕ(x, a) does not fork over any model M ⊇ A. Some p ∈ Sϕ (C) containing ϕ(x, a) does not divide over A.

Proof. The equivalence of 1, 2 and 3 follows directly from Proposition 8.3 applied to the empty type ð(x). 2 ⇒ 4 is a consequence of point 2 of Remark 4.4. 4 ⇒ 1 follows from point 7 of Remark 4.2. 5 ⇒ 1 is clear. 3 ⇒ 5. Assume p ∈ Sϕ (C) is as in 3. We check that p does not divide over A. Let ϕ(x, a1 ), . . . , ϕ(x, an ), ¬ϕ(x, b1 ), . . . , ¬ϕ(x, bk ) ∈ p and let ø(x; y1 , . . . , yn , z1 , . . . , zk ) be the formula ϕ(x, y1 ) ∧ · · · ∧ ϕ(x, yn ) ∧ ¬ϕ(x, z1 ) ∧ · · · ∧ ¬ϕ(x, zk ). Notice that ø(x; y1 , . . . , yn , z1 , . . . , zk ) is stable. Let q(x) = {ø(x, c) : p(x) ` ø(x, c)} ∪ {¬ø(x, c) : p(x) ` ¬ø(x, c)}. Then q ∈ Sø (C) is definable over acleq (A) and ø(x; a1 , . . . , an , b1 , . . . , bk ) ∈ q. By the equivalence 1 ⇔ 3 applied to ø(x; a1 , . . . , an , b1 , . . . , bk ), ø(x; a1 , . . . , an , b1 , . . . , bk ) does not divide over A. Hence p does not divide over A. a Proposition 8.5. Let T be simple and let ∆ = {ϕi (x, yi ) : i < n} be a set of stable formulas. The following are equivalent for any ∆-type ð(x). 1. ð(x) does not fork over A. 2. ð(x) is finitely satisfiable in every model M ⊇ A. 3. Some p(x) ∈ S∆ (C) containing ð(x) is definable over acleq (A). Proof. Recall that if each ϕi (x, yi ) is stable, then any boolean combination ø(x; y1 , . . . , yn ) of the formulas ϕi (x, yi ) is stable. Since dividing and forking are the same in a simple theory, 1 ⇔ 2 follows from Corollary 8.4.

50

8. Stable forking

1 ⇒ 3. If ð(x) does not fork over A, it can be extended to some p(x) ∈ S∆ (C) that does not fork over A. By 1 ⇔ 2, p(x) is finitely satisfiable in every model M ⊇ A and by Proposition 8.1 it is definable over every model M ⊇ A. This clearly implies that it is definable over acleq (A). 3 ⇒ 2. By Proposition 8.1, p is finitely satisfiable in every M ⊇ A and hence the same is true for ð(x). a Corollary 8.6. Let T be stable, p(x) ∈ S(A) and M ⊆ A (possibly A = C). The following are equivalent: 1. p(x) does not fork over M . 2. p(x) inherits from M . 3. p(x) coinherits from M . 4. p(x) is M -definable. 5. Some p ∈ S(C) extending p does not split over M . Proof. The equivalence between points 2, 3, and 4 follows from Corollary 7.14. The equivalence of 4 and 5 follows from Proposition 7.12. The equivalence of 1 and 3 follows from Corollary 8.4. a Corollary 8.7. Let T be stable and let p(x) ∈ S(C). The following are equivalent: 1. p(x) does not fork over A. 2. p(x) coinherits from every M ⊇ A. 3. p(x) inherits from every M ⊇ A. 4. p(x) does not split over any M ⊇ A. 5. p(x) is definable over every M ⊇ A. 6. p(x) is definable over acleq (A). 7. The Cantor–Bendixson rank of p  ϕ in Sϕ (C) is CBϕ (p  A) for all ϕ. 8. p(x) has a bounded orbit in Aut(C/A) (in fact of size ≤ 2|T | if it is finitary). Proof. The equivalence between points 2, 3, 4, 5, and 6 follows from Corollary 8.6 (for 6 observe that p is definable over acleq (A) if and only if it is definable over every model M ⊇ A). 1 ⇔ 6 follows from Corollary 8.4. 6 ⇔ 7 follows from Corollary 6.15. 7 ⇒ 8. The orbit of p  ϕ is bounded by Mltϕ (p  A) < ù and hence the orbit of p is bounded by 2|T | . 8 ⇒ 6. Let cϕ be the canonical parameter of the definition of p  ϕ. Since cϕ has bounded orbit in Aut(C/A), in fact it has finite orbit. Hence cϕ ∈ acleq (A) and p  ϕ is definable over acleq (A). a Corollary 8.8. Let T be stable, p(x) ∈ S(A) and ϕ(x, y) ∈ L. The following are equivalent: 1. p(x) ∪ {ϕ(x, a)} does not fork over A. 2. CBø (p(x) ∪ {ϕ(x, a)}) = CBø (p) for all ø. 3. CBϕ (p(x) ∪ {ϕ(x, a)}) = CBϕ (p).

8. Stable forking

51

Proof. 1 ⇒ 2. Let p(x) ∈ S(C) be an extension of p(x) ∪ {ϕ(x, a)} which does not fork over A. By Corollary 8.7, CBø (p(x)) is the Cantor–Bendixson rank of p(x)  ø in Sø (C). Hence CBø (p(x) ∪ {ϕ(x, a)}) = CBø (p(x)). 2 ⇒ 3 is obvious. We prove 3 ⇒ 1. Let p(x) ∈ S(C) be a nonforking extension of p(x). By Corollary 8.7, CBϕ (p(x)) is the Cantor–Bendixson rank of p(x)  ϕ. Let q(x) ∈ S(C) be such that q(x)  ϕ is consistent with p(x) ∪ {ϕ(x, a)} and has Cantor–Bendixson rank CBϕ (p(x) ∪ {ϕ(x, a)}). By corollaries 6.14 and 6.15 p(x)  ϕ and q(x)  ϕ are A-conjugate. Since ϕ(x, a) ∈ q(x)  ϕ, p(x) ∪ {ϕ(x, a)} is contained in an A-conjugate of p(x), a global type which does not fork over A. Hence p(x) ∪ {ϕ(x, a)} does not fork over A. a Corollary 8.9. Let T be stable, A ⊆ B and p(x) ∈ S(B). The following are equivalent: 1. p(x) does not fork over A. 2. CBϕ (p) = CBϕ (p  A) for all ϕ. Proof. It is an immediate consequence of Corollary 8.8. a Proposition 8.10. Let T be simple. If ϕ(x, y) ∈ L is stable, for every A, a there is some ó(x) ∈ L(A) equivalent to a positive boolean combination of A-conjugates of ϕ(x, a) and such that for every p(x) ∈ S(A), ó(x) ∈ p(x) if and only if p(x) ∪ {ϕ(x, a)} does not fork over A. Proof. Apply Proposition 8.3 with p(x) = ð(x). a Corollary 8.11 (Open mapping theorem). Let T be stable and let A ⊆ B. The set NF(B, A) of all p(x) ∈ S(B) which do not fork over A is closed in S(B) and the restriction mapping p 7→ p  A from NF(B, A) onto S(A) is open. Proof. The restriction map from S(C) onto S(B) is continuous and hence closed and the image of NF(C, A) is NF(B, A). Hence it is enough to check that NF(C, A) is closed. Now, \ NF(C, A) = {p ∈ S(C) : p coinherits from M } M ⊇A

and for each M , {p ∈ S(C) : p coinherits from M } is closed since it is the closure of {tp(a/C) : a ∈ M }. The fact that the restriction map from NF(B, A) onto S(A) is open is an immediate consequence of Proposition 8.10. a Corollary 8.12. If T is stable, any nonforking extension of a nonisolated type is nonisolated. Proof. By Proposition 8.10 or Corollary 8.11. a Corollary 8.13. If T is simple, any nonforking extension of a type which is not isolated by stable formulas is also not isolated by stable formulas. Proof. By Proposition 8.10. a

Chapter 9

LASCAR STRONG TYPES

Here we will consider relations R, meaning binary relations between tuples of C of length α for some ordinal α. The ordinal α usually is intended to be a natural number, but we do not impose restrictions. Definition 9.1. A relation R is bounded if for some cardinal κ there is no sequence (ai : i < κ) such that ¬R(ai , aj ) for all i < j < κ. The relation is finite if this bound κ is in fact a natural number. Observe that for definable relations finiteness is equivalent to boundedness. Note also that bounded relations are always reflexive. Remark 9.2. For every cardinal ë, any intersection of ë bounded relations is a bounded relation. Proof. Let (Rl : l < ë) be a sequence of bounded relations. For all l < ë let κl be a bound for Rl and let κ = ë + sup{κl : l < ë}. Assume there T are (ai : i < (2κ )+ ) such that¬R(ai , aj ) for all i < j < (2κ )+ , where R = l<ë Rl . ˝ By Erdos–Rado ((2κ )+ → (κ + )2κ ) for some l < ë there is a subset I ⊆ (2κ )+ + of cardinality κ such that ¬Rl (ai , aj ) for all i < j in I . This contradicts the choice of κl . a Lemma 9.3. 1. The number of A-invariant relations on tuples of length α |T |+|A|+|α| is at most 22 . 2. There is a smallest A-invariant bounded equivalence relation (among tuples of a fixed length). Proof. 1. By Lemma 1.2 the number of sets of types over A of tuples of length α is an upper bound. 2 follows from Remark 9.2. a Definition 9.4. We say that the tuples a, b have the same Lascar strong type Ls over A and we write a ≡A b if a and b are equivalent in the least A-invariant bounded equivalence relation. The Lascar strong type of a over A can be Ls defined as the equivalence class Lstp(a/A) of a in ≡A . Hence Ls

a ≡A b ⇔ Lstp(a/A) = Lstp(b/A). In the case A = ∅ we omit the subscript. 53

54

9. Lascar strong types

Definition 9.5. Let x, y be finite tuples of variables of the same length. We say that the formula è(x, y) is thick if it defines a relation which is finite and symmetric. Note that all thick formulas define reflexive relations. For any set A and for any tuples of variables x, y of the same length, the set of all thick formulas over A in (finite subtuples of) the variables x, y will be ncA (x, y). In the case A = ∅ we omit the subscript. Let ncn (x, y) be the n-times composition of nc(x, y); more formally: for n ≥ 2, ncnA (x, y) is the type ∃y1 . . . yn−1 (ncA (x, y1 ) ∧ ncA (y1 , y2 ) ∧ · · · ∧ ncA (yn−1 , y)) and we complete the definition taking nc1A (x, y) = ncA (x, y) and taking as nc0A (x, y) the equation x = y. Remark 9.6. 1. The conjunction and the disjunction of thick formulas are thick formulas. 2. Any consequence of a thick formula is a finite formula. 3. If ϕ(x, y) is finite, then ϕ(x, y) ∧ ϕ(y, x) is thick. Proof. It is an easy exercise. a Lemma 9.7. For any a, b, |= ncA (a, b) if and only if a, b start an infinite A-indiscernible sequence. Proof. If a, b start an infinite A-indiscernible sequence, then |= è(a, b) for every thick formula è(x, y) over A. Now assume |= ncA (a, b). Let p(x, y) = tp(ab/A). By Ramsey’s Theorem and compactness, to prove that a, b start an infinite A-indiscernible sequence it is enough to check that there is an infinite sequence (ai : i < ù) such that |= p(ai , aj ) for all i < j < ù. For this we have to prove for any ϕ ∈ p, the consistency of {ϕ(xi , xj ) : i < j < ù}. If this set of formulas is inconsistent, then ¬ϕ(x, y) is finite and therefore (¬ϕ(x, y) ∧ ¬ϕ(y, x)) ∈ ncA (x, y). Hence |= ¬ϕ(a, b), a contradiction. a Proposition 9.8. ncA (x, a) does not divide over A. Proof. If (ai : iS< ù) is A-indiscernible and a = a0 , then by Lemma 9.7, a is a realization of i<ù ncA (x, ai ). a Ls

Proposition 9.9. The relation ≡A of equality of Lascar strong type over A is the transitive closure of the relation of starting an A-indiscernible sequence. W Hence it is defined by the infinite disjunction n ncnA (x, y). Proof. Since the relation of starting an infinite indiscernible sequence is defined by the type ncA (x, y) consisting of finite formulas, it is bounded. Hence its transitive closure E is also bounded. Since E is a bounded ALs invariant equivalence relation, ≡A ⊆ E. For the other direction it suffices Ls to show that if a, b start an infinite A-indiscernible sequence then a ≡A b. Ls Let κ be a strict upper bound for the number of ≡A -classes. Choose an

55

9. Lascar strong types Ls

A-indiscernible sequence of length κ starting with a, b. If a ≡ 6 A b then by 0 Ls 0 0 0 A-invariance a ≡ 6 A b for any a , b in the sequence, which contradicts the choice of κ. a Definition 9.10. For a, b tuples of the same length, we put dA (a, b) = 0 if a = b. If a 6= b and there is some n < ù for which there are infinite A-indiscernible sequences I1 , . . . , In and tuples a1 , . . . , an−1 such that a = a1 , b = an and for each i < n, ai , ai+1 ∈ Ii , then we define dA (a, b) as the least such n. If there is no such n we put dA (a, b) = ∞. The diameter over A of a class X of tuples of the same length is diamA (X ) = sup{dA (a, b) : a, b ∈ X }. Remark 9.11. 1. dA (a, b) ≤ n if and only if |= ncnA (a, b). Ls 2. a ≡A b ⇔ dA (a, b) < ∞. a

Proof. Clear by definition and Proposition 9.9.

Lemma 9.12. 1. If |= ncA (a, b), then there is a model M ⊇ A such that a ≡M b. 2. If a ≡M b for some model M ⊇ A, then |= nc2A (a, b). Proof. 1. Fix an infinite A-indiscernible sequence I starting with a, b. By Corollary 1.7 I is indiscernible over some model M ⊇ A. Then a ≡M b. 2. Assume that a ≡M b for some model M ⊇ A. We show that |= ∃z(è(a, z) ∧ è(b, z)) for every thick formula è(x, y) over A. Let n be the maximal length of a sequence a1 , . . . , an such that |= ¬è(ai , aj ) for all i < j ≤ n. We can find such a1 , . . . , an in M . For some i ≤ n, |= è(a, ai ). Since a ≡M b also |= è(b, ai ). a Proposition 9.13. Equality of Lascar strong types over A is the transitive closure of the relation of having the same type over a model containing A. a

Proof. Clear by Proposition 9.9 and Lemma 9.12.

Definition 9.14. The group Autf(C/A) of strong automorphisms over A of the monster model C is the subgroup of Aut(C/A) generated by the automorphisms fixing a small submodel containing A: [ Autf(C/A) = h Aut(C/M )i. M ⊇A Ls

Corollary 9.15. a ≡A b if and only if f(a) = b for some f ∈ Autf(C/A). a

Proof. It follows from Proposition 9.13. Ls

Ls

Corollary 9.16. If a ≡A b then for any c there is some d such that ac ≡A bd . Proof. Choose f ∈ Autf(C/A) such that f(a) = b and put d = f(c).

a

56

9. Lascar strong types

Definition 9.17. Let ð(x) be a partial type over A such that for each a, b |= ð there is some f ∈ Aut(C) fixing setwise ð(C) such that f(a) = b. Let U be a class consisting of tuples of the length of x. We say that U is c-free over ð if for some n < Snù there are automorphisms f1 , . . . , fn fixing setwise ð(C) such that ð(C) ⊆ i=1 fi (U ). U is called weakly c-free over ð if for some V non c-free over ð, the union UV is c-free over ð. When ð is fixed, we will sometimes omit to mention it. A formula is (weakly) c-free if it defines a (weakly) c-free class. A type is (weakly) c-free if it only implies (weakly) c-free formulas. Remark 9.18. Let ð(x) be a partial type as in the definition of c-freeness. 1. If U is (weakly) c-free and U ⊆ V, V is also (weakly) c-free. 2. I = {U : U is not weakly c-free} is an ideal (i.e. it is downward closed and closed under finite unions). 3. Any partial type over B ⊇ A which is weakly c-free can be extended to a complete weakly c-free p(x) ∈ S(B). 4. ð(x) is a c-free type. Proof. For 3 note that every filter F such that I ∩ F = ∅ can be extended to an ultrafilter F 0 such that I ∩ F 0 = ∅. a Lemma 9.19. Let ð(x) be a partial type as in the definition of c-freeness. Let U be definable. If U is weakly c-free, then for some definable V , V is not c-free and UV is c-free. Proof. To fix notation assume U ⊆ Cα . There is some non c-free V such that UV is c-free and hence Sn there are automorphisms Tn f1 , . . . , fn fixing setwise ð(C) such that ð(C) ⊆ i=1 fi (UV ). Let W = i=1 Cα r fi (U ). It is definable. If fn+1 is the identity, then ð(C) ⊆ W ∪(Cα rW ) = W ∪

n [ i=1

fi (U ) ⊆ UW ∪

n [ i=1

fi (UW ) =

n+1 [

fi (UW )

i=1

Sn and therefore UW is c-free. Since i=1 fi (V ) is not c-free and W ∩ ð(C) ⊆ Sn f (V ), W ∩ð(C) is not c-free either. This implies that W is not c-free. a i=1 i Proposition 9.20. Let ð(x) be a type over A as in the definition of c-freeness and let ϕ(x, y) ∈ L. Assume T is simple. 1. If ϕ(x, a) is weakly c-free over ð, then ð(x) ∪ {ϕ(x, a)} does not fork over A. 2. If ϕ(x, y) is stable, ð(x) is a complete type over A, and ð(x) ∪ {ϕ(x, a)} does not fork over A, then ϕ(x, a) is c-free over ð. Proof. 1. Let ϕ(x, a) be weakly c-free (over ð). By Lemma 9.19 for some ø(x, z) ∈ L, for some b, ø(x, b) is not c-free and ϕ(x, a) ∨ ø(x, b) is c-free. There are automorphisms f1 , . . . , fk fixing setwise ð(C) such that ð(x) ` Wk ϕ 0 (x, a 0 ) ∨ ø 0 (x, b 0 ), where ϕ 0 (x, a 0 ) = i=1 ϕ(x, fi (a)) and ø 0 (x, b 0 ) = Wk 0 0 i=1 ø(x, fi (b)). Note that ø (x, b ) is not c-free. Assume ð(x) ∪ {ϕ(x, a)}

9. Lascar strong types

57

forks over A. Then ð(x) ∪ {ϕ 0 (x, a 0 )} forks (and divides) over A too. For some finite conjunction è(x) of formulas of ð, for some sequence (ai0 : i < ù) of realizations of tp(a 0 /A), for some l < ù, {è(x) ∧ ϕ 0 (x, ai0 ) : i < ù} is l -inconsistent. Let g1 , . . . , gl be automorphisms fixing A pointwise and Wl sending a 0 to a10 , . . . , al0 respectively. Then ð(x) ` i=1 ø 0 (x, gi (b 0 )), which implies that ø 0 (x, b 0 ) is c-free, a contradiction. 2. Let ð(x) = p(x) ∈ S(A) and assume p(x) ∪ {ϕ(x, a)} does not fork over A. By Proposition 8.10 there is a formula ó(x) ∈ p which is equivalent to a positive boolean combination of A-conjugates of ϕ(x, a), say Wk to i=1 øi (x, ai ) where each øi (x, ai ) is a conjunction of A-conjugates of ϕ(x, a). If we select an A-conjugate ϕ(x, ai0 ) of ϕ(x, a) in each øi (x, ai ) we Wk see that p(x) ` i=1 ϕ(x, ai0 ), which shows that ϕ(x, a) is c-free over p. a Proposition 9.21. Let ð(x) be a partial type over A as in the definition of c-freeness. Consider the sets 1. P = {q(x, y) ∈ S(∅) : q(x, y) ∪ ð(x) ∪ ð(y) is consistent}, 2. Pw = {q(x, y) ∈ P : q(a, y) is weakly c-free for some (all ) a |= ð}. Then P and Pw are nonempty closed subsets of S(∅). Moreover, if S ⊆ Pw is nonempty and relatively open, then for some k < ù there are c1 , . . . , ck |= ð such that for all b |= ð there is some d |= ð such that tp(bd ) ∈ S and tp(ci d ) ∈ S for some i. Proof. With respect to the first statement, only the case of Pw needs some verification. Fix a |= ð. Note that ð(y) is weakly c-free and therefore it can be extended to a complete weakly c-free type q(a, y). Thus Pw 6= ∅. For closedness, note that Pw = {q(x, y) ∈ S(∅) : q(x, y) ⊇ Γ(x, y)} if Γ(x, y) = {¬ϕ(x, y) ∈ L : ϕ(a, y) is not weakly c-free}. Recall from chapter 1 that for any set of formulas Σ(x, y) over ∅, [Σ] denotes the corresponding closed set in S(∅). Now choose ϕ(x, y) ∈ L witnessing that S is not empty and it is relatively open in Pw . Making S smaller if necessary, we can assume that S = Pw ∩ [ϕ(x, y)]. Fix c |= ð. Note that ϕ(c, y) is weakly c-free. By Lemma 9.19, there is a non c-free ø(e, y) such that ϕ(c, y) ∨ ø(e, y) is c-free. Hence for some automorphisms f1 , . . . , fk fixing setwise ð(C), (∗) ð(y) `

k _

ϕ(ci , y) ∨ ø(ei , y)

i=1

where ci = fi (c) and ei = fi (e). Now, since S ⊆ S(∅) is closed, S = [ñ(x, y)] S for some partial type ñ(x, y) over ∅. Note that ñ(c, C) = q∈S q(c, C). Claim 1. (ϕ(c, C) ∪ ø(e, C)) r ñ(c, C) is not c-free.

58

9. Lascar strong types

Assume the claim is true and our proposition is false for k, for c1 , . . . , ck and for some b |= ð. We will reach a contradiction. For all d ∈ ñ(b, C), Sk Sk d 6∈ i=1 ñ(ci , C), and by (∗), d ∈ i=1 ϕ(ci , C) ∪ ø(ei , C) and therefore ñ(b, C) ⊆

k [

((ϕ(ci , C) ∪ ø(ei , C)) r ñ(ci , C)).

i=1

Choose an automorphism fk+1 fixing setwise ð(C) and such that fk+1 (c) = b. Let ck+1 = fk+1 (c) and ek+1 = fk+1 (e). Since ϕ(ck+1 , C) ∪ ø(ek+1
, C) clearly is a subset of ñ(ck+1 , C) ∪ (ϕ(ck+1 , C) ∪ ø(ek+1 , C)) r ñ(ck+1 , C) , it follows that k+1 [ ϕ(ck+1 , C) ∪ ø(ek+1 , C) ⊆ (ϕ(ci , C) ∪ ø(ei , C)) r ñ(ci , C). i=1

Sk+1

Since ϕ(c, C) ∪ ø(e, C) is c-free, i=1 (ϕ(ci , C) ∪ ø(ei , C)) r ñ(ci , C) must be c-free and this contradicts the claim. Proof of Claim 1. Assume (ϕ(c, C) ∪ ø(c, C)) r ñ(c, C) is c-free. There are automorphisms g1 , . . . , gn fixing setwise ð(C) such that (with notation ci0 = gi (c) and ei0 = gi (e)), ð(C) ⊆

n [

(ϕ(ci0 , C) ∪ ø(ei0 , C)) r ñ(ci0 , C)

i=1

Vn


that is, ð(y) ∧ i=1 (¬ϕ(ci0 , y) ∧ ¬ø(ei0 , y)) ∨ ñ(ci0 , y) is inconsistent. By compactness, there are
that αi (x, y) ` Vn αi (x, y) ∈ ñ(x, y) for i = 1, . . . , n such ϕ(x, y) and ð(y) ∧ i=1 (¬ϕ(ci0 , y) ∧ ¬ø(ei0 , y)) ∨ αi (ci0 , y) is inconsistent. Vn Let α(x, y) = i=1 αi (x, y). Then also ð(y) ∧

n ^


(¬ϕ(ci0 , y) ∧ ¬ø(ei0 , y)) ∨ α(ci0 , y)

i=1

Sn is inconsistent, that is ð(C) ⊆ i=1 (ϕ(ci0 , C) ∪ ø(ei0 , C)) r α(ci0 , C), which implies that (ϕ(c, y) ∨ ø(e, y)) ∧ ¬α(c, y) is c-free. Then also (ϕ(c, y) ∧ ¬α(c, y))∨ø(e, y) is c-free and (ϕ(c, y)∧¬α(c, y)) is weakly c-free and can be extended to some weakly c-free complete type q(c, y). This is a contradiction, because q(x, y) ∈ Pw and ϕ(x, y) ∈ q(x, y), which implies q(x, y) ∈ Pw ∩ [ϕ(x, y)] = S = [ñ(x, y)] ⊆ [α(x, y)]. a a Theorem 9.22. A type-definable Lascar strong type has finite diameter. Proof. Let ð(x) be a partial type defining Lstp(a/A) over aA. Then ð(x) satisfies our assumptions in the definition of c-freeness. Let P, Pw be as in Proposition 9.21. Let Xn = P ∩ [ncn (x, y)]. Then, by Proposition 9.9, S P = n∈ù Xn . Since Pw is a nonempty closed set, by the Baire Category

59

9. Lascar strong types

Theorem, some Xn has nonempty interior in Pw . By the Proposition 9.21, there is some k < ù and c1 , . . . , ck |= ð such that for all a, b |= ð there are c, d |= ð such that tp(a, c), tp(b, d ) ∈ Xn and tp(ci , c), tp(cj , d ) ∈ Xn for some i, j ≤ k. Let m = max{d (cl , cl 0 ) : 1 ≤ l, l 0 ≤ k}. Then d (a, b) ≤ d (a, c) + d (c, ci ) + d (ci , cj ) + d (cj , d ) + d (d, b) ≤ 4n + m. a Corollary 9.23. 1. If for each n < ù there is a Lascar strong type over A of diameter ≥ n, then there is a Lascar strong type over A which is not type-definable. 2. Equality of Lascar strong types over A is type-definable if and only if for some n < ù it is defined by ncnA (x, y). Proof. 1. For each n < ù fix a tuple an whose Lascar strong type over A has diameter at least n and consider a = (an : n < ù). It is easy to check that Lstp(a/A) has infinite diameter. By Theorem 9.22, it is not type-definable. Ls 2. Assume the relation ≡A of equality of Lascar strong types is typedefinable. By 1 there is a bound n < ù for the diameter of any Lascar strong type over A. Since dA (a, b) ≤ n is equivalent to |= ncnA (a, b), the type Ls ncnA (x, y) defines ≡A . a Definition 9.24. As in the case of A-invariance, there is a smallest typedefinable over A bounded equivalence relation (on tuples of a given length). We say that the tuples a, b have the same Kim–Pillay strong type over A or the KP same bounded type over A and we write a ≡A b if a and b are equivalent in the least type-definable over A bounded equivalence relation. s We say that a, b have the same strong type over A and we write a ≡A b if a and b are equivalent in every A-definable finite equivalence relation. As usual, in case A = ∅ we omit the subscript. Ls

KP

Remark 9.25. 1. If a ≡A b, then a ≡A b. KP s 2. If a ≡A b, then a ≡A b. s 3. If a ≡A b, then a ≡A b. Proof. 1 is clear since every equivalence relation that is type-definable over A is A-invariant. Similarly for 2, since every A-definable finite equivalence relation is bounded and type-definable over A. For 3 observe that for each ϕ(x) ∈ L(A), the equivalence relation E(x, y) defined by (ϕ(x) ↔ ϕ(y)) is A-definable and has only two classes. a Definition 9.26. The strong type of a over A is defined by stp(a/A) = tp(a/acleq (A)). s

Lemma 9.27. stp(a/A) = stp(b/A) if and only if a ≡A b.

60

9. Lascar strong types

Proof. Assume stp(a/A) = stp(b/A). Let E be a finite A-definable equivalence relation, say defined by ϕ(x, y, c) where c ∈ A and ϕ(x, y, z) ∈ L. Let ø(z) ∈ tp(c) be a formula expressing that ϕ(x, y, z) defines an equivalence relation in x, y and consider the relation F (ux; vy) defined by F (ux; vy) ⇔ (¬ø(u) ∧ ¬ø(v)) ∨ (ø(u) ∧ u = v ∧ ϕ(x, y, u)). It is a 0-definable equivalence relation and therefore the equivalence classes [ca]F and [cb]F are imaginary elements. Since F (cx; cy) defines E and E is finite, these imaginaries are algebraic over A, that is, they are elements of acleq (A). This clearly implies [ca]F = [cb]F and therefore E(a, b). For the other direction, notice that according to Proposition 1.11 a relation R defined by a formula ϕ(x) ∈ acleq (A) has finitely many A-conjugates and it is therefore a union of classes of some finite A-definable equivalence relation. a s

Proposition 9.28. Let T be stable. If a ≡A b, A ⊆ B, a ^ | A B and b ^ | A B, s

then a ≡B b. Proof. Let p(x) = stp(a/A) = stp(b/A), let p ∈ S(C) be a nonforking extension of stp(a/B) and let q ∈ S(C) be a nonforking extension of stp(b/B). Since a ^ | A B, a ^ | A acleq (B) and therefore p does not fork over A. By Corollary 8.7 p is definable over acleq (A). By the same argument q is definable over acleq (A) and by Proposition 6.13 p = q. Hence stp(a/B) = stp(b/B). a Ls

s

Corollary 9.29. If T is stable, then ≡A = ≡A for every A. s

Proof. Let a ≡A b. Choose M ⊇ A such that M ^ | A ab. Then a ^ | AM and b ^ | A M . By Proposition 9.28 a ≡M b and hence by Lemma 9.12 and Ls

Proposition 9.9, a ≡A b.

a

Theorem 9.30 (Finite equivalence relation theorem). Let T be a stable theory. Let A ⊆ B, r(x) ∈ S(A), and let p(x), q(x) ∈ S(B) be two different nonforking extensions of r. Then for some ϕ(x) ∈ L(B) equivalent to a formula over acleq (A), ϕ ∈ p while ¬ϕ ∈ q. There is also a finite A-definable equivalence relation E such that p(x) ∪ q(y) ` ¬E(x, y). Proof. Let p (x) ∈ S(B ∪ acleq (A)) be an extension of p. If p 0 (x)  acleq (A) ∪ q(x) is consistent then there is some extension q 0 (x) ∈ S(B ∪ acleq (A)) of q such that p 0  acleq (A) = q 0  acleq (A). But then p0 and q 0 are different nonforking extensions of the same strong type, which contradicts Corollary 9.29. Hence p0 (x)  acleq (A) ∪ q(x) is inconsistent and there is some ø(x) ∈ p 0 (x)  acleq (A) such that q(x) ` ¬ø(x). Let ϕ(x) be the disjunction of all B-conjugates of ø. Then p(x) ` ϕ(x), q(x) ` ¬ϕ(x) and ϕ(x) ∈ L(acleq (A)) is equivalent to a formula over B. 0

9. Lascar strong types

61

With respect to the last assertion, by Proposition 1.11 ϕ(x) defines a union of classes of a finite A-definable equivalence relation E, and then clearly p(x) ∪ q(y) ` ¬E(x, y). a

Chapter 10

THE INDEPENDENCE THEOREM

Lemma 10.1. Let T be simple and let κ > |T |. If (ai : i < κ) is Aindependent, then for any set B of cardinality < κ there is some i < κ such that B^ | A ai . Proof. By choice of κ, there is a proper subset X ⊆ κ such that B ^ | Aa {ai : X i < κ} where aX = (ai : i ∈ X ). Take i 6∈ X . Then B ^ | Aa ai and, by X Corollary 5.21, ai ^ | A aX . By symmetry and transitivity, B ^ | A ai . a Lemma 10.2. Let T be simple. For any a, A and B ⊇ A there is some a 0 such Ls that a 0 ≡A a and a 0 ^ | A B. Proof. Let κ be a cardinal larger than |T | + |B|. Let (ai : i < κ) be a Morley sequence in tp(a/A) starting with a0 = a. By Lemma 10.1 there is Ls some i < κ such that B ^ | A ai . Clearly, a ≡A ai . a Ls

Lemma 10.3. Let T be simple and let a ≡A b. For any c, B there is some d Ls such that ac ≡A bd and d ^ | Ab B. Ls

Proof. By Corollary 9.16 there is some d 0 such that ac ≡A bd 0 and by Corollary 9.15, there is a strong automorphism f ∈ Autf(C/A) such that Ls f(ac) = bd 0 . By Lemma 10.2 there is some d such that d ≡Ab d 0 and d^ | Ab B. Again by Corollary 9.15 there is some g ∈ Autf(C/Ab) such that g(d 0 ) = d . It follows that g ◦ f ∈ Autf(C/A) and g ◦ f(ac) = bd . Hence Ls ac ≡A bd . a Lemma 10.4. Let T be simple. If (ai : i < ù+ù) is an infinite A-indiscernible sequence, then (ai : ù ≤ i < ù + ù) is a Morley sequence over A{ai : i < ù}. Proof. Let a = (ai : i < ù). Clearly (ai : ù ≤ i < ù + ù) is Aaindiscernible. It suffices to show that it is Aa-independent. Let X be a finite subset of {i : ù ≤ i < ù + ù} an let i < ù + ù be larger than every element in X . By symmetry it will be enough to check that aX ^ | Aa ai , where aX = (aj : j ∈ X ). But this is clear since by A-indiscernibility tp(aX /Aaai ) is finitely satisfiable in a. a 63

64

10. The independence theorem

Proposition 10.5. Let T be simple and let ð(x, y) be a set of formulas over A. If (ai : i ∈ I ) is an A-indiscernible sequence and ð(x, ai ) does not fork over A S for some (every) i ∈ I , then i∈I ð(x, ai ) does not fork over A. Proof. We may assume I = ù and ð(x, a0 ) does not fork over A. Let us first assume that (ai : i < ù) S is a Morley sequence over A. Since ð(x, a0 ) does not divide over A, i<ù ð(x, ai ) is consistent. Let n < ù and let Φ(x, y0 , . . . , yn−1 ) = ð(x, y1 ) ∪ · · · ∪ ð(x, yn ). We will show that Φ(x, a0 , . . . , an−1 ) does not divide over A. If bi = an·i . .S . an·i+n−1 , then (bi : i < ù) is an infinite Morley sequence in tp(b0 /A) and i<ù Φ(x, bi ) is consistent. By Proposition 5.15, Φ(x, b0 ) does not divide over A. Now let us consider the general case, where (ai : i < ù) is just an Aindiscernible sequence. Choose b = (bi : i < ù) such that (bi : i < ù)a (ai : i < ù) is A-indiscernible. By Lemma 10.4 (ai : i < ù) is a Morley sequence over A ∪ b. Let p(x, y) ∈ S(Ab) be such that p(x, a0 ) extends ð(x, a0 ) and does S not fork over A. Then it does not fork overSAb and by the first case, i<ù p(x, ai ) does not fork over Ab. Let c |= i<ù p(x, ai ) be such that c^ | Ab (ai : i < ù). Since p(x, a0 ) does not fork over A, also c ^ | A ba0 . Hence S c^ | A b(ai : i < ù), which shows that i<ù ð(x, ai ) does not fork over A. a Lemma 10.6. Assume a, b start an infinite A-indiscernible sequence and c^ | Aa b. Then for some d , the extended tuples ac, bd also start an infinite A-indiscernible sequence. Proof. We may assume A = ∅. Let c ^ | a b and assume (ai : i < ù) is an infinite indiscernible sequence with a = a0 and b = a1 . Since (an : n ≥ 1) is aindiscernible and c ^ | a b, by Lemma 4.7 there is an ac-indiscernible sequence 0 (an : n ≥ 1) such that (an : n ≥ 1) ≡ab (an0 : n ≥ 1). Thus we may assume that an = an0 for all n ≥ 1. Let c0 = c and choose for n ≥ 1 some cn such that ca0 a1 . . . ≡ cn an an+1 . . . . Since (an : n ≥ 1) is ac-indiscernible, cab ≡ caam . Hence cab ≡ cn an an+m , i.e., in the sequence (cn an : n < ù) all triangles cn an an+m have the same type p(x, y, z) = tp(cab). By Ramsey’s Theorem there is an indiscernible sequence (dn bn : n < ù) where all triangles dn bn bn+m satisfy p(x, y, z). Clearly we may assume that c = d0 , a = b0 and b = b1 . Take d = d1 . a Proposition 10.7. Let T be simple, let ϕ(x, y), ø(x, z) ∈ L(A), and assume that ϕ(x, a) ∧ ø(x, b) does not fork over A. If b, b 0 start an infinite A-indiscernible sequence and a ^ | Ab b 0 , then ϕ(x, a) ∧ ø(x, b 0 ) does not fork over A. Proof. Apply Lemma 10.6 finding a 0 such that ba, b 0 a 0 start an infinite A-indiscernible sequence. By Proposition 10.5, ϕ(x, a) ∧ ø(x, b) ∧ ϕ(x, a 0 ) ∧ ø(x, b 0 ) does not fork over A. In particular ϕ(x, a) ∧ ø(x, b 0 ) does not fork over A. a

65

10. The independence theorem

Corollary 10.8. Let T be simple, let ϕ(x, y), ø(x, z) ∈ L(A), and assume Ls that ϕ(x, a) ∧ ø(x, b) does not fork over A. If b ≡A b 0 , a ^ | A b, and a ^ | A b0, 0 then ϕ(x, a) ∧ ø(x, b ) does not fork over A. Proof. Note that we can assume a ^ | A bb 0 since by Lemma 10.2 we can Ls

replace b 0 by some b 00 ^ | Aa b such that b 00 ≡Aa b 0 , which implies b 00 b ^ | A a. Find b1 , . . . , bn such that b = b1 , b 0 = bn and bi , bi+1 start an infinite Aindiscernible sequence. Let a 0 be such that a 0 ≡Abb 0 a and a 0 ^ | Abb 0 b1 , . . . , bn . By Proposition 10.7 we see that ϕ(x, a 0 ) ∧ ø(x, bi ) does not fork over A for all i ≤ n. Hence ϕ(x, a) ∧ ø(x, b 0 ) does not fork over A. a Corollary 10.9 (Independence Theorem). Let T be a simple theory, let ϕ(x, y), ø(x, z) ∈ L(A), and assume a ^ | A b. If there are c, d such that Ls

|= ϕ(c, a), c ^ | a, |= ø(d, b), d ^ | b, and c ≡A d, A

A

then ϕ(x, a) ∧ ø(x, b) does not fork over A. Ls

Proof. Using Lemma 10.3, choose b 0 ^ | Ac ab such that cb 0 ≡A db. Then |= ϕ(c, a) ∧ ø(c, b 0 ) and c ^ | A ab 0 . Therefore ϕ(x, a) ∧ ø(x, b 0 ) does not fork 0 over A. Since a ^ | A bb by Corollary 10.8, ϕ(x, a) ∧ ø(x, b) does not fork over A. a Corollary 10.10. Let T be simple. 1. Assume A is a common subset of B and C . Assume B ^ | A C , and let Ls

b^ | A B, c ^ | A C , be such that b ≡A c. Then for some d ^ | A BC , d ≡B b and d ≡C c. 2. Let (ai : i ∈ I ) be an A-independent sequence, let ði (x) a partial type over Aai which does not fork over A and assume that whenever (bi : i ∈ I ) Ls is S a sequence of realizations bi |= ði then bi ≡A bj for all i, j ∈ I . Then i∈I ði (x) does not fork over A. 3. Let (ai : i ∈ I ) be an M -independent sequence, let ði (x) a partial type over S Mai which does not fork over M and extends p(x) ∈ S(M ). Then i∈I ði (x) does not fork over M . Proof. 1 follows from Corollary 10.9. For 2 we may assume I = ù and then using 1 it is easy to prove by induction that ð0 (x) ∪ · · · ∪ ðn (x) does not fork over A for all n < ù. 3 follows from 2 since bi ≡M bj implies Ls bi ≡M bj . a Ls

Proposition 10.11. Let T be simple. If a ≡A b and a ^ | A b, then a, b start a Morley sequence (ai : i < ù) over A. Proof. Let p = tp(ab/A). We prove first that for any cardinal κ there is an A-independent sequence (ai : i < κ) such that |= p(ai , aj ) for all i < j < κ.

66

10. The independence theorem Ls

Note that this implies ai ≡A aj . The sequence is constructed inductively startS ing with a0 = a and a1 = b. We choose as ai a realization of j
fork over A. Note that c ≡A d whenever c |= p(aj , x) and d |= p(aj 0 , x). Therefore it is clear that we can apply the generalized version of the Independence Theorem stated in point 2 of Corollary 10.10 to obtain the desired result. Now, once we have this A-independent sequence we still need to make it A-indiscernible. But this can be done easily by Proposition 1.6. a Ls

Proposition 10.12. If T is simple, then a ≡A b if and only if there is some c such that a, c start an infinite A-indiscernible sequence and b, c start an infinite indiscernible sequence over A. Ls

Ls

Proof. Assume a ≡A b and find with Lemma 10.2 some c such that c ≡A a and c ^ | A ab. By Proposition 10.11 a, c start an infinite Morley sequence over A and b, c start an infinite Morley sequence over A. a Ls

Corollary 10.13. If T is simple, then the relation ≡A of equality of Lascar strong type over A is type-definable over A by nc2A (x, y). Proof. Clear, by Proposition 10.12. a Ls

Corollary 10.14. If T is simple, then for any set A, a ≡A b if and only if Ls a ≡A0 b for all finite A0 ⊆ A. Proof. By Corollary 10.13. a Ls

Definition 10.15. T is G-compact over A if the relation ≡A of equality of Lascar strong type is type-definable for every possible length of tuples. Remark 10.16. The following are equivalent: 1. T is G-compact over A. Ls KP 2. ≡A =≡A . Ls 3. For some n < ù, ≡A is defined by the type ncnA (x, y). 4. For some n < ù, all Lascar strong types over A have diameter ≤ n. Proof. Clearly both 2 and 3 imply 1. By Remark 9.11, 3 is equivalent to 4. Ls For 1 ⇒ 2 note that if ≡A is type-definable then it is the least type-definable KP over A bounded equivalence relation and hence it must be ≡A . Finally for 1 ⇒ 3 use Corollary 9.22. a Corollary 10.17. If T is simple, then T is G-compact over A for every A. Ls

Proof. By Corollary 10.13, ≡A is type-definable over A.

a

Example 10.18. The first example of a non G-compact theory was discovered by Ziegler. It is presented in [10]. Proposition 10.19. T is G-compact over A if and only if the two following conditions hold :

10. The independence theorem Ls

67

1. For every n < ù the relation ≡A of equality of Lascar strong type on n-tuples is type-definable. 2. Autf(C/A) is closed in the topology of Aut(C/A). Proof. Assume T is G-compact over A. We need only to check 2. Let f ∈ Aut(C/A) be an accumulation point of Autf(C/A) and let m be a tuple enumerating a model M ⊇ A. For each finite subtuple a of m there is some g ∈ Autf(C/A) such that g(a) = f(a) and hence by Corollary 9.15, Ls Ls Ls a ≡A f(a). Since ≡A is type-definable over A, it follows m ≡A f(m). Again by Corollary 9.15, there is some g ∈ Autf(C/A) such that f(m) = g(m). Then g −1 ◦ f is the identity on M and therefore it belongs to Autf(C/A). It follows that f = g ◦ g −1 ◦ f ∈ Autf(C/A). Ls Assume now 1 and 2. We check that for each infinite ordinal α, ≡A as a relation on tuples of length α is type-definable. Fix α-tuples of variables x = (xi : i < α) and y = (yi : i < α) and for each finite I ⊆ α choose a type Ls ΣI (xI , yI ) over A defining ≡A on the corresponding finite tuples of variables xI = (xi : i ∈ I ) and yI = (yi : i ∈ I ). Let Σα (x, y) be the union of all these types. It is easy to see that |= Σα (a, b) if and only if the corresponding Ls elementary mapping a 7→ b preserves ≡A on finite subtuples. It is also clear that these elementary mappings can be extended by back-and-forth. Hence, if |= Σα (a, b) then a 7→ b can be extended to some f ∈ Aut(C/A) such that for Ls every finite tuple c, c ≡A f(c). This means that f is an accumulation point Ls of Autf(C/A). By 2, f ∈ Autf(C/A) and therefore a ≡A b. a

Chapter 11

CANONICAL BASES

Definition 11.1. The multiplicity of a type p(x) ∈ S(A) is the number Mlt(p) of its global nonforking extensions p(x) ∈ S(C). If there is a proper class of global nonforking extensions of p, we say that p has unbounded multiplicity and we write Mlt(p) = ∞; otherwise we say that p has bounded multiplicity. A stationary type is a type of multiplicity 1. Thus over any B ⊇ A a stationary type p(x) ∈ S(A) has a unique nonforking extension q(x) ∈ S(B). We use the notation p|B for q. Lemma 11.2. Let T be simple. If p ∈ S(A) is stationary, then its global nonforking extension is definable over A. Proof. Let p be the global nonforking extension of p, and let ϕ(x, y) ∈ L. We will show that p  ϕ is A-definable. Let ∆ϕ (y) and ∆¬ϕ (y) be types over A given by Corollary 5.23 for p and ϕ and for p and ¬ϕ respectively. By compactness, the conjunction ø(y) of a finite subset of ∆ϕ (y) is inconsistent with ∆¬ϕ (y). It is clear that ø(y) defines p  ϕ. a Corollary 11.3. Let T be simple. If types over models are stationary, then T is stable. Proof. Lemma 11.2 implies that in this situation every global type is definable. a Proposition 11.4. Let T be simple. 1. If p ∈ S(M ) has bounded multiplicity, then p is stationary. 2. If p ∈ S(A) has bounded multiplicity, then every extension of p over acleq (A) is stationary. Proof. 1. Assume p ∈ S(M ) has two nonforking extensions over A ⊇ M , say p1 and p2 . We will show that no nonforking extension of p is stationary. This implies that p has an unbounded number of nonforking global extensions. Let q be a nonforking extension of p over B ⊇ M . To show that q is not stationary we may assume B ^ | M A. By the Independence Theorem (Corollary 10.10) applied to p1 and q we obtain a type q1 ∈ S(AB) extending q ∪ p1 which does not fork over M . Similarly, by applying it to p2 and q we obtain a type q2 ∈ S(AB) extending q ∪ p2 which does not fork over M . Then 69

70

11. Canonical bases

q1 , q2 are two different nonforking extensions of q over AB, which shows that q is not stationary. 2. Let p 0 (x) ∈ S(acleq (A)) be a (nonforking) extension of p and let M ⊇ A. Any nonforking extension of p0 over M has bounded multiplicity and by point 1 it is stationary. We show that p 0 has only one nonforking extension over M . This will ensure the stationarity of p 0 . Let q1 ∈ S(M ) be a nonforking extension of p 0 . By Lemma 11.2 the global nonforking extension of q1 is M definable. Since p 0 has bounded multiplicity, this global nonforking extension has a bounded number of acleq (A)-conjugates and therefore it is definable over acleq (A). Therefore q1 is definable over acleq (A). Now assume that q2 ∈ S(M ) is another nonforking extension of p 0 . Again, q2 is stationary and definable over acleq (A). Let d1 xϕ(x, y) ∈ L(acleq (A)) and d2 xϕ(x, y) ∈ L(acleq (A)) be definitions of q1 and q2 respectively. We will show that if ϕ(x, a) ∈ q1 then ϕ(x, a) ∈ q2 . Let bi |= qi . Then bi ^ | A M and |= ϕ(b1 , a). Let r(y) = stp(a/A) and let ∆(x) be the partial type over acleq (A) given by Corollary 5.23 for r(y) and ϕ −1 (y, x). Then |= ∆(b1 ). Since it is a partial type over acleq (A), also |= ∆(b2 ) and therefore there is some a 0 such that a 0 ^ | A b2 , a 0 |= r(y) and |= ϕ(b2 , a 0 ). We may find this a 0 with the additional property that a 0 ^ | Ab M . In this case 2 0 0 a ^ | M b2 and hence by stationarity ϕ(x, a ) belongs to the global nonforking extension of q2 , that is, |= d2 xϕ(x, a 0 ). Since this formula is over acleq (A) and s a ≡A a 0 we conclude that |= d2 xϕ(x, a), that is, ϕ(x, a) ∈ q2 . a Remark 11.5. Let T be stable. 1. Any strong type is stationary. 2. Any type over a model is stationary. Proof. Clear by Proposition 9.28.

a

Lemma 11.6. Let T be simple. tp(ab/A) is stationary if and only if tp(b/A) and tp(a/Ab) are stationary. Proof. Let tp(ab/A) be stationary. We check first that tp(b/A) is stationary. For this assume B ⊇ A, b 0 ≡A b, b ^ | A B, and b 0 ^ | A B. Choose a0 such that a0 b ≡A ab and a0 ^ | Ab B and then choose a1 such that a1 b 0 ≡A ab and a1 ^ | Ab 0 B. Then a0 b ^ | A B and a1 b 0 ^ | A B. Since tp(ab/A) is stationary, 0 a0 b ≡B a1 b and therefore b ≡B b 0 . Now we check that tp(a/Ab) is stationary. Assume a ≡Ab a 0 , B ⊇ A, a ^ | Ab B and a 0 ^ | Ab B. Let b 0 ≡A b be such that b0 ^ | A B. We have just proved that tp(b/A) is stationary. Hence b ≡B b 0 . Choose now a0 , a1 such that a0 a1 b 0 ≡B aa 0 b. Then a0 b 0 ^ | A B, a1 b 0 ^ | A B, 0 0 0 0 and a0 b ≡A a1 b ≡A ab. By stationarity a0 b ≡B a1 b , and hence ab ≡B a 0 b and a ≡Bb a 0 . Now assume tp(b/A) and tp(a/Ab) are stationary. Let ab ≡A a 0 b 0 , B ⊇ A, ab ^ | A B, and a 0 b 0 ^ | A B. By stationarity of tp(b/A), b ≡B b 0 . Now choose

11. Canonical bases

71

a0 such that a0 b ≡B a 0 b 0 . Then a0 ^ | Ab B and a0 ≡Ab a. By stationarity of tp(a/Ab), a0 ≡Bb a. Hence ab ≡B a0 b ≡B a 0 b 0 . a Remark 11.7. If T is stable, then any two global nonforking extensions of p(x) ∈ S(A) are A-conjugate. Proof. Let p1 , p2 ∈ S(C) be two nonforking extensions of p and let pi = pi  acleq (A). There is some f ∈ Aut(C/A) such that p1f = p2 . By Remark 11.5, p2 is stationary. Since pf1 and p2 are nonforking extensions of p2 , they coincide. a Proposition 11.8. Let T be stable and let p(x) ∈ S(A) be finitary. 1. Mlt(p) ≤ 2|T | . 2. If Mlt(p) ≥ ù, then Mlt(p) ≥ 2ù . Proof. 1. Choose some B ⊆ A of cardinality ≤ |T | such that p does not fork over B. Since every nonforking extension of p is a nonforking extension of p  B, it is enough to check that Mlt(p  B) ≤ 2|T | . Let M ⊇ B be a model of cardinality ≤ |T |. By Remark 11.5 every type over M extending p  B is stationary. Then Mlt(p  B) is bounded by the number of extensions of p  B over M and this number is ≤ |S(M )| ≤ 2|T | . 2. By Corollary 8.11, the set of nonforking extensions over C of p(x) ∈ S(A) is a closed set in S(C). By Remark 11.7 any two points in this set are connected by a homeomorphism induced by an automorphism of C over A. Hence if this set has an isolated point, any other point is isolated and therefore it is finite. If it does not have isolated points, it is a nonempty perfect set and therefore it contains at least ≥ 2ù points. a Definition 11.9. Two stationary types p(x) ∈ S(A), q(x) ∈ S(B) are called parallel if they have a common nonforking extension. We then write p k q. Note that q = (p|AB)  B. Definition 11.10. Let p ∈ S(C) be definable. A subset B of Ceq is a canonical base of p if for every f ∈ Aut(C), pf = p if and only if f fixes B pointwise. Clearly, p is definable over A if and only if B ⊆ dcleq (A). Hence, in a sense, a canonical base of p is a smallest set over which p is definable. Remark 11.11. 1. If p is definable and B is a canonical base of p, then p is definable over B. 2. If B, B 0 are canonical bases of the definable type p, then dcleq (B) = dcleq (B 0 ). 3. For every definable global type p, if cϕ is the canonical parameter of some definition of p  ϕ, then (cϕ : ϕ ∈ L) is a canonical base of p. Proof. Easy exercise. a Definition 11.12. Let p(x) ∈ S(A) be a stationary type in a simple theory. We call B a canonical base of p if B is a canonical base of the (definable)

72

11. Canonical bases

global nonforking extension of p. We use the notation Cb(p) for dcleq (B) where B is a canonical base of p. Finally, in a stable theory we define Cb(a/A) = Cb(stp(a/A)). Remark 11.13. Let T be simple. A set B is a canonical base of the stationary type p ∈ S(A) if and only if for each f ∈ Aut(C): p k p f if and only if f fixes B pointwise. Proof. Clear, since an automorphism f fixes the global nonforking extension of p if and only if p k p f. a Proposition 11.14. Let T be simple. If p(x) ∈ S(A) is stationary, then Cb(p) ⊆ dcleq (A). If T is stable, then Cb(a/A) ⊆ acleq (A). Proof. If f ∈ Aut(C/A) and p(x) ∈ S(A) is stationary, then pf = p k p and therefore f fixes Cb(p) pointwise. Hence Cb(p) ⊆ dcleq (A). Now note that in a stable theory Cb(a/A) = Cb(stp(a/A)) and stp(a/A) = tp(a/acleq (A)) is stationary. a Proposition 11.15. Let T be stable. Let B be a canonical base of p(x) ∈ S(C). Then p does not fork over A if and only if B ⊆ acleq (A). Moreover the following are equivalent: 1. p is definable over A. 2. B ⊆ dcleq (A). 3. p does not fork over A and p  A is stationary. Proof. If p does not fork over A then p(x) = p  acleq A is stationary and has B as a canonical base. Hence by Proposition 11.14 B ⊆ acleq (A). On the other hand if B ⊆ acleq A then p is definable over acleq (A) and hence it does not fork over A. Equivalence between 1 and 2 is immediate. Now we prove the equivalence with 3. If p is definable over A, then by Corollary 8.7 p does not fork over A. Moreover p is the only element of its orbit in Aut(C/A) and then, by Remark 11.7, p  A is stationary. For the other direction, if p does not fork over A and p  A is stationary, then clearly p is the only element of its orbit in Aut(C/A) and therefore it is A-definable. a Proposition 11.16. Let T be stable. If A ⊆ B, the following are equivalent. 1. a ^ | A B. 2. Cb(a/B) ⊆ acleq (A). 3. Cb(a/A) = Cb(a/B). Proof. 1 ⇔ 2 follows from Proposition 11.15. As for 3, note that if a ^ | AB then stp(a/A) and stp(a/B) have the same global nonforking extension and therefore Cb(a/A) = Cb(a/B). On the other hand, if their canonical bases coincide, then Cb(a/B) = Cb(a/A) ⊆ acleq (A). a

11. Canonical bases

73

Lemma 11.17. Let T be stable. If a is a tuple definable over the tuple b, then Cb(a/A) ⊆ Cb(b/A). Hence, two interdefinable tuples have the same canonical base over any set. Proof. Let C = Cb(b/A). Since b ^ | C A and tp(b/C ) is stationary then a^ | C A and by Lemma 11.6 tp(a/C ) is stationary. Hence Cb(a/A) ⊆ C . a Lemma 11.18. Let T be simple. Let p(x), q(y) ∈ S(A) and assume one of them is stationary. Let a, a 0 be realizations of p and let b, b 0 be realizations of q. If a ^ | A b and a 0 ^ | A b 0 then ab ≡A a 0 b 0 . If p and q are stationary, then tp(ab/A) is also stationary. Proof. Without loss of generality, q is stationary. Choose c such that ab ≡A a 0 c. Then c ≡A b 0 , c ^ | A a 0 and b 0 ^ | A a 0 . Since q is stationary, c ≡Aa 0 b 0 . Then a 0 b 0 ≡A a 0 c ≡A ab. The last assertion follows from Lemma 11.6. a Lemma 11.19. Let T be simple. Let (I, <) be a linearly ordered set and for each i ∈ I , let pi (xi ) ∈ S(A) be stationary. Let (ai : i ∈ I ) be an A-independent sequence where ai |= pi for all i ∈ I . If (bi : i ∈ I ) is an A-independent sequence such that bi |= pi for all i ∈ I , then (ai : i ∈ I ) ≡A (bi : i ∈ I ). Moreover tp((ai : i ∈ I )/A) is stationary. Proof. We can assume I is finite and then it can be proved easily by induction on |I | using Lemma 11.18. a Definition 11.20. Assume T is simple. Let pi (xi ) ∈ S(A) for each i ∈ I and assume each of the types pi is stationary. The product of the types (pi : i ∈ I ) is the stationary type tp((ai : i ∈ I )/A) where (ai : i ∈ I ) is A-independent and ai |= pi . By Lemma 11.19 it is well defined. We denote N it by i∈I pi . In the finite case we use the notation p1 ⊗ · · · ⊗ pn . If all the types pi are equal to p(x) ∈ S(A), the notations are p I and p n . Remark 11.21. Let T be simple. If (ai : i < α) is an A-independent sequence of realizations of the stationary type p(x) ∈ S(A), then it is a Morley sequence in p and tp((ai : i < α)/A) = p α . Hence, if (bi : i < α) is another A-independent sequence of realizations of p, then (ai : i < α) ≡A (bi : i < α). Proof. A-indiscernibility of (ai : i < α) can be justified noticing that for every n < ù, for every i0 < · · · < in < α, tp(a0 , . . . , an /A) = tp(ai0 , . . . , ain /A) = p n+1 .

a

Lemma 11.22. Let ϕ(x, y) ∈ L stable, let p ∈ S(C) and assume p  ϕ is M -definable. If cϕ is the canonical parameter of some definition of p  ϕ over M , then cϕ ∈ dcleq (ai : i < ù) for some Morley sequence (ai : i < ù) in p  M . Proof. By Proposition 8.2 p  ϕ is definable over some Morley sequence (ai : i < ù) in p  M . a

74

11. Canonical bases

Proposition 11.23. If T is stable, then for each Morley sequence (ai : i < ù) in stp(a/A), Cb(a/A) ⊆ dcleq (ai : i < ù). Proof. Let p be the global nonforking extension of p(x) = stp(a/A) and fix some ϕ(x, y) ∈ L and some model M ⊇ A. Let cϕ be the canonical parameter of a definition of p  ϕ. By Lemma 11.22 cϕ ∈ dcleq (bi : i < ù) for some Morley sequence (bi : i < ù) in p  M . Note that (bi : i < ù) s is also a Morley sequence in stp(a/A). By Remark 11.21 (ai : i < ù) ≡A (bi : i < ù) and therefore there is some f ∈ Aut(C/acleq (A)) sending each bi to ai . Since p is definable over acleq (A), cϕ ∈ acleq (A). It follows that cϕ ∈ dcleq (ai : i < ù). Since Cb(a/A) is definable over (cϕ : ϕ ∈ L), we conclude that Cb(a/A) ⊆ dcleq (ai : i < ù). a

Chapter 12

ABSTRACT INDEPENDENCE RELATIONS

Notation 12.1. In this chapter ^ | will be an arbitrary ternary relation bef tween sets. We will use ^ | for the nonforking independence relation as defined in 5.1. Definition 12.2. An independence relationis a ternary relation ^ | between sets satisfying the following axioms: 1. Invariance. If A ^ | C B and f ∈ Aut(C), then f(A) ^ | f(C ) f(B). 0 0 2. Monotonicity. If A ^ | C B, A ⊆ A, and B ⊆ B, then A0 ^ | C B 0. 3. Right base monotonicity. If A ^ | C B and C ⊆ D ⊆ B, then A ^ | D B. 4. Left transitivity. If B ⊆ C ⊆ D, C ^ | B A, and D ^ | C A, then D ^ | B A. 5. Left normality. If A ^ | C B, then AC ^ | C B. 6. Extension. If A ^ | C B and B 0 ⊇ B, then A0 ^ | C B 0 for some A0 ≡BC A. 7. Left finite character. If A0 ^ | C B for all finite A0 ⊆ A, then A ^ | C B. 8. Local character. For every A there is a cardinal number κ(A) such that for any B there is some C ⊆ B such that |C | < κ(A) and A ^ | C B. We say that the independence relation ^ | is strict if it additionally satisfies 9. Anti-reflexivity. If A ^ | C A, then A ⊆ acl(C ). For a tuple a, a ^ | C B means that A ^ | C B where A is the set enumerated by a. Similarly for other notations like a ^ | C b, etc. Remark 12.3. Note that the property of right normality if A ^ | B then A ^ | BC C

C

follows from extension and invariance. In this context, right base monotonicity may be reformulated as: if A ^ | B and D ⊆ B, then A ^ | B. C

CD

Similarly, under left normality, left transitivity can be reformulated as: if C ^ | A, and D ^ | A, then CD ^ | A. B

BC

75

B

76

12. Abstract independence relations

In this form the property is one direction of the Pairs Lemma: CD ^ | A if and only if C ^ | A and D ^ | A. B

B

BC

Note also that right base monotonicity and weak local character give what is known as the existence property: A^ | B. B

Proposition 12.4. Assume ^ | satisfies the first five axioms of independence relations and also the extension property. If a ^ | C B, then there is a BC indiscernible sequence (ai : i < ù) such that ai ≡BC a and a 0, (ai0 : 0 < i < n) ^ | a00 . C

We prove it by induction on n. It is clear for n = 1. By the induction hypothesis and left normality, C (ai0 : 0 < i < n) ^ | C a00 . By construction of the sequence 0 and right base monotonicity, an ^ | C (a 0 :0
C (ai0 : 0 < i ≤ n) ^ | C (a 0 :0 κ(B). We can extend our sequence to a BC -indiscernible sequence (ai : i < κ). By finite character and invariance, a
77

12. Abstract independence relations

by monotonicity, B ^ | Ca aj . By left normality BCa<j ^ | Ca aj and also <j <j Ca<j ^ | C aj . By left transitivity, BCa<j ^ | C aj . By monotonicity B ^ | C aj . Since a ≡BC aj , by invariance B ^ | C a. a Corollary 12.6. Any independence relation is symmetric, that is: if A^ | C B, then B ^ | C A. Proof. It is an immediate consequence of propositions 12.4 and 12.5. a ∗

Definition 12.7. For any invariant ternary relation ^ | we define ^ | as ∗ follows: A ^ | C B if and only if for all B 0 ⊇ B there is some A0 ≡BC A such that A0 ^ | C B 0 . Sometimes it is more convenient to state it in terms of automorphisms: for all B 0 ⊇ B there is some f ∈ Aut(C/BC ) such that f(A) ^ | C B 0. ∗

∗

Remark 12.8. For any invariant ^ | ,^ | is also invariant and A ^ | C B implies A^ | C B. ∗

Proposition 12.9. For any monotone invariant ^ | ,^ | has the extension property. ∗ Proof. Let A ^ | C B and B ⊆ B 0 . Let a enumerate A and let x be a corresponding tuple of variables. We claim that there is a type p(x) ∈ S(CB 0 ) extending tp(a/CB) and such that for each cardinal κ there is a κ-saturated model M ⊇ CB 0 and some a 0 |= p such that a 0 ^ | C M . Assume not, and fix 0 for each p(x) ∈ S(CB ) extending tp(a/CB) a corresponding cardinal κp for which there is no κp -saturated model M ⊇ CB 0 with a realization a 0 |= p such that a 0 ^ | C M . Let κ be the supremum of all these cardinals κp and choose ∗ a κ-saturated model M ⊇ CB 0 . Since a ^ | C B, there is some a 0 ≡CB a such 0 0 0 that a ^ | C M . Then tp(a /CB ) provides a contradiction. Now using the claim we fix some p(x) ∈ S(CB 0 ) as indicated. Let a 0 |= p. ∗ ∗ We will show that a 0 ^ | C B 0 . This will establish the extension property for ^ | . 00 0 00 0 00 Let B ⊇ B . We need to show that for some a ≡CB 0 a (i.e., some a |= p), a 00 ^ | C B 00 . Let κ = |C ∪B 0 |+ +|B 00 |+ù and by the claim choose a κ-saturated M ⊇ CB 0 and some a 00 |= p such that a 00 ^ | C M . By κ-saturation there is 0 an automorphism f ∈ Aut(C/CB ) such that f(B 00 ) ⊆ M . By monotonicity a 00 ^ | C f(B 00 ). By invariance f −1 (a 00 ) ^ | C B 00 . Since f −1 (a 00 ) |= p we have finished. a Remark 12.10. Each one of the properties of monotonicity, right base monotonicity, left transitivity, left normality, and anti-reflexivity is preserved when ∗ passing from ^ | to ^ | . Proposition 12.11. Assume ^ | satisfies the first five axioms of independence ∗ ∗ and also left finite character. If ^ | satisfies local character, then ^ | is an independence relation.

78

12. Abstract independence relations

Proof. By Remark 12.10 and Proposition 12.9 we only need to show that ∗ ∗ | | is symmetric. Note that ^ ∗ has left finite character. But first we check that ^ | satisfies the hypotheses of Proposition 12.4 and ^ | satisfies the hypotheses ^ ∗ of Proposition 12.5. Hence A ^ | C B implies B ^ | C A. ∗ ∗ Now assume A ^ | C B and let us prove that B ^ | C A. Let A0 ⊇ A. Since ∗ A0 ^ | AC AC , by extension there is some f ∈ Aut(C/AC ) such that ∗ ∗ ∗ f(A0 ) ^ | AC ACB. By monotonicity f(A0 ) ^ | AC B. Since A ^ | C B, by left ∗ ∗ transitivity and monotonicity of ^ | , f(A0 ) ^ | C B. Hence B ^ | C f(A0 ) and ∗ −1 0 f (B) ^ | C A , which shows that B ^ | C A. ∗ ∗ Now we check left finite character of ^ | . Assume a ^ | C B for all finite ∗ tuples a ∈ A. To prove that A ^ | C B, consider some B 0 ⊇ B. By existence ∗ and extension, there is some f ∈ Aut(C/BC ) such that f(A) ^ | CB B 0 . Hence ∗ ∗ A^ | CB f −1 (B 0 ). By symmetry f −1 (B 0 ) ^ | BC A. For each finite tuple a ∈ A, ∗ ∗ we have a ^ | C B and a ^ | BC f −1 (B 0 ). By symmetry and left transitivity we ∗ −1 then obtain a ^ | C f (B 0 ) for all finite tuples a ∈ A. Hence a ^ | C f −1 (B 0 ) for all finite tuples a ∈ A. By left finite character of ^ | , A^ | C f −1 (B 0 ). By 0 invariance f(A) ^ | CB. a ∗

Proposition 12.12. Let ^ | be invariant and monotone. Then ^ | =^ | if and only if ^ | has the extension property. Proof. One direction follows from Proposition 12.9. The other direction ∗ ∗ is clear by definition of ^ | since ^ | is stronger than ^ | . a f

Definition 12.13. It has already been mentioned that ^ | is nonforking d independence. We define ^ | as nondividing independence. To be precise: d

1. A ^ | C B if and only if for every tuple a ∈ A, tp(a/BC ) does not divide over C . f 2. A ^ | C B if and only if for every tuple a ∈ A, tp(a/BC ) does not fork over C . d

f

Proposition 12.14. (^ | )∗ = ^ | . f

Proof. By Remark 4.4 we know that ^ | has the extension property. Since f

d

f

d

| implies ^ | , it follows that ^ | implies (^ | )∗ . For the other direction, ^ d f assume A(^ | )∗C B but A ^ 6 | C B. For some tuple a ∈ A, for some formula ϕ(x, y) ∈ L, for some b ∈ BC , |= ϕ(a, b) and ϕ(x, b) forks over C . Then for some ø1 (x, y1 ), . . . , øn (x, yn ) ∈ L, for some b1 , . . . , bn , |= ϕ(x, b) → ø1 (x, b1 ) ∨ · · · ∨ øn (x, bn )

12. Abstract independence relations

79

and each ø(x, bi ) divides over C . Let B 0 = Bb1 , . . . , bn . By assumption there d is some a 0 ≡BC a such that a 0 ^ | C B 0 . Since |= ϕ(a 0 , b), |= øi (a 0 , bi ) for some i. This implies that tp(a 0 /B 0 ) divides over C , a contradiction. a d

Remark 12.15. The relation ^ | has the properties of invariance, monotonicity, right base monotonicity, left transitivity, left normality, and anti-reflexivity. f d Therefore ^ | satisfies all these properties and extension. Moreover ^ | and f

| satisfy left and right finite character. ^ Proof. For left transitivity see Proposition 4.8 and for anti-reflexivity see point 5 in Remark 4.2. The other properties are straightforward. a Proposition 12.16. The following are equivalent. 1. T is simple. f 2. ^ | satisfies local character. d 3. ^ | satisfies local character. f

4. ^ | is an independence relation. d 5. ^ | is an independence relation. Proof. We know that simplicity of T implies all the other conditions. It is clear that 4 implies 2 and that 5 implies 3. It is also clear that 2 implies 3. We now check that simplicity follows from 3. Assume T is not simple. Then for some p(x) ∈ S(∅) for some ϕ(x, y) ∈ L, for some k < ù, D(p(x), ϕ, k) = ∞. d The cardinal κ(a) given by local character of ^ | is clearly the same for any realization a of p. Let κ be regular and larger than this cardinal. By Proposition 3.11 there is a sequence (ai : i < κ) such that p(x) ∪ {ϕ(x, ai ) : i < κ} is consistent and for each i < κ, ϕ(x, ai ) k-divides over a
80

12. Abstract independence relations f

nonforking independence ^ | in simple theories. By Corollary 12.6 we know that ^ | is symmetric. Therefore ^ | is also right transitive, that is, if B ⊆ C ⊆ D, A ^ | C and A ^ | D, then A ^ | D B

C

B

and has right finite character, that is, if A ^ | B0 for all finite B0 ⊆ B, then A ^ | B. C

C

If I is a linearly ordered set, we say that (ai : i ∈ I ) is ^ | -independent over C when ai ^ | C a
C

C

then there is some c such that c^ | AB, c ≡A a, and c ≡B b. C

In other terms, if C ⊆ A ∩ B and A ^ | C B, for any two types p(x) ∈ S(A) and q(x) ∈ S(B) which are ^ | -free over C and have a common restriction to C , their union can be extended to a complete type over AB which is ^ | -free over C . d

Proposition 12.19. ^ | is stronger than any independence relation ^ | , that d is: if A ^ | C B, then A ^ | C B. d

Proof. Assume a ^ | C b but a ^ 6 | C b. Let κ(a) be the cardinal given for a by the local character property and choose a regular cardinal κ > κ(a). Since b ^ | C C , there is a ^ | -Morley sequence (bi : i < κ) over C starting with b0 = b. Its initial segment (bi : i < ù) can be obtained as in Proposition 12.4 (using freely the symmetry of ^ | ) and for its extension to a sequence of length κ we need only to preserve C -indiscernibility since ^ | -independence over C is granted by invariance and finite character. Now let p(x, y) = tp(ab/C ). S Since p(x, b) does not divide over C , i<κ p(x, bi ) is consistent. Let a 0 be a realization of this union of types. Then a 0 bi ≡C ab for all i < κ, which implies that a 0 ^ 6 | C bi for all i < κ. If a 0 ^ | Cb bi then (by transitivity) a 0 ^ | C bi , which
12. Abstract independence relations

81

Lemma 12.20. Let ^ | be an independence relation. Assume ^ | satisfies the Independence Theorem over C . Then for any p(x, y) ∈ S(C ), if (ai : i < α) is an ^ | -independent sequence over C and each p(x, ai ) is a ^ | -free extension of S its common restriction to C , then i<α p(x, ai ) is ^ | -free over C . Proof. We inductively S construct a chain of types (qi : i < α) such that qi (x) ∈ S(Ca
f

Proof. If T is simple then clearly ^ | =^ | is an independence relation (see Proposition 12.16) and satisfies the Independence Theorem over models (see Corollary 10.10). For the opposite direction, by Proposition 12.19 we d d know that ^ | ⊆ ^ | . We will show now that ^ | ⊆ ^ | . From this it d

d

will follow that ^ | =^ | and hence that ^ | has local character in T . By Proposition 12.16 T is simple. d Let a ^ | C b. We check that a ^ | C b. Let p(x, y) = tp(ab/C ) and let S (bi : i < ù) be C -indiscernible with b0 = b. We will show that i<ù p(x, bi ) is consistent. Let κ(b) be the cardinal number given for b by the local character property and choose a regular cardinal κ > κ(b). Extend the given sequence to a C -indiscernible sequence (bi : i ≤ κ). By Corollary 1.7 there is a model M ⊇ C such that (bi : i ≤ κ) is M -indiscernible. Starting with M0 = M it is easy now to construct a chain of models (Mi : i < κ) such that Cb
82

12. Abstract independence relations S 0 fact S over C ). By Lemma 12.20, i<ù p (x, bi ) is consistent. In particular a i<ù p(x, bi ) is consistent. Theorem 12.22. T is stable if and only if there is an independence relation | ^ in T which satisfies one of the two equivalent conditions: 1. Types over models are ^ | -stationary, that is, for any p(x) ∈ S(M ), for any B ⊇ M there is a unique ^ | -free extension of p over B. 2. Every type has a bounded number of ^ | -free extensions, that is, for each tuple of variables x there is a cardinal ì such that for every p(x) ∈ S(A) for every B ⊇ A there are at most ì ^ | -free extensions of p over B. d

Moreover if T is stable and ^ | is as indicated, then ^ | =^ | . d f Proof. If T is stable, T is simple and ^ | = ^ | is an independence relation. Moreover (see Remark 11.5 and Proposition 11.8) conditions 1 and 2 hold. 1 implies 2. Let α be the length of x and let κ be the cardinal given by local character according to Remark 12.17. Let ì = 2|T |+κ . We want to show that ì is an upper bound for the number of ^ | -free extensions of p(x) ∈ S(A) over any other larger set. For this we may assume that |A| ≤ κ because there is some C ⊆ A of cardinality < κ such that p is ^ | -free over C and then a bound for p  C is also a bound for p. There is a model M ⊇ A of cardinality κ. The number of extensions of p to a complete type over M is bounded by |S(M )| ≤ 2|T |+κ = ì. Since every type over M is stationary, the number of | -free extensions of p over any set is also bounded by ì. ^ d

2 implies stability of T and ^ | =^ | (and hence it implies 1). Fix ì as in 2 and fix an n-tuple of variables x. Choose κ > |T | witnessing the local character of ^ | for n as in Remark 12.17. Choose ë ≥ ì such that ë = ë<κ . We show that T is stable in ë. Let |A| ≤ ë. For each p(x) ∈ S(A) there is some C ⊆ A such that p is ^ | -free over C and |C | < κ. There are ≤ ë<κ = ë such subsets C ⊆ A, over each such C there are ≤ 2|T |+|C | ≤ ë<κ = ë types q(x) ∈ S(C ) and for each q(x) ∈ S(C ) there are at most ì ≤ ë ^ | free extensions of q over A. The number of types p(x) ∈ S(A) is therefore bounded by ë. Thus, T is stable. d d By Proposition 12.19 we know that ^ | ⊆^ | . To check ^ | ⊆^ | assume p(x) = tp(a/BC ) divides over C . Every global extension p ∈ S(C) of p forks over C and therefore (see Corollary 8.7) has unboundedly many C -conjugates. But if p is ^ | -free over C , then over any larger set p has an extension which is ^ | -free over C and hence the number of its C -conjugates is bounded by ì. Therefore a ^ 6 | C B. a Proposition 12.23. The following are equivalent. 1. T is not simple.

83

12. Abstract independence relations

2. For some ϕ(x, y) ∈ L, some k < ù, there is an indiscernible sequence (ci ai : i < ù) such that for all i < ù, |= ϕ(ci , a0 ) and ϕ(x, ai ) k-divides over {cj aj : j < i}. 3. For some ϕ(x, y) ∈ L, some k < ù, there are a tuple c and some cindiscernible sequence (ai : i < ù) such that for all i < ù, |= ϕ(c, ai ) and ϕ(x, ai ) k-divides over a
4. ^ | is right transitive. f 5. ^ | is right transitive. d

Proof. By Proposition 5.18 and by the fact that in a simple theory ^ | = f

| , conditions 2 and 3 follow from 1. ^

84

12. Abstract independence relations d

f

2 ⇒ 4 and 3 ⇒ 5. Since ^ | and ^ | are left transitive, it is clear that symmetry implies they are right transitive. 4 ⇒ 1 and 5 ⇒ 1. Fix an ordered set of order type ù + 2 + ù ∗ , where ù ∗ is the reverse order of ù, say 0 < 1 < · · · < ù < ù + 1 < · · · < −2 < −1. Assume T is not simple. By Proposition 12.23 and compactness there is some ϕ(x, y) ∈ L and some k for which there is an indiscernible sequence (ci ai : i ∈ ù + 2 + ù ∗ ) such that for each i, ϕ(x, ai ) k-divides over {cj aj : j < i} and for all j ≤ i, |= ϕ(ci , aj ). Let I = {ci : i ∈ ù} and let J = {ci : i ∈ ù ∗ }. d Since |= ϕ(cù+1 , aù ), cù+1 ^ 6 | I Jaù . Since tp(cù+1 /IJ ) is finitely satisfiable in f

f

I , cù+1 ^ | I J . Since tp(cù+1 /aù IJ ) is finitely satisfiable in J , cù+1 ^ | IJ aù . d

f

This contradicts the right transitivity of ^ | and ^ | .

a

Chapter 13

SUPERSIMPLE THEORIES

Definition 13.1. T is supersimple if for every finitary p ∈ S(A) there is a finite A0 ⊆ A such that p does not fork over A0 . In other words, for any finite tuple a, for any set A, there is a finite A0 ⊆ A such that a ^ | A A. By 0 Proposition 4.13 this implies T is simple. T is superstable if it is stable and supersimple. Definition 13.2. κ(T ) is the least cardinal ì such that for each finite tuple a, for each set A there is some B ⊆ A such that |B| < ì and a ^ | B A. If there is no such cardinal ì we set κ(T ) = ∞. Remark 13.3. 1. T is simple if and only if κ(T ) < ∞ if and only if κ(T ) ≤ |T |+ . 2. T is supersimple if and only if κ(T ) = ù. Proof. For 1 use Proposition 4.13. a Proposition 13.4. The following are equivalent. 1. T is supersimple. 2. There is no infinite sequence (ϕi (x, ai ) : i < ù) such that {ϕi (x, ai ) : i < ù} is consistent and for each i < ù, ϕi (x, ai ) divides ( forks) over a
86

13. Supersimple theories

As usual, SU(p) = ∞ if SU(p) ≥ α for all α, and SU(p) = α if SU(p) ≥ α but SU(p) 6≥ α + 1. U is defined by the same conditions for 0 and for a limit number α. For a successor ordinal the rule is as follows: • For p(x) ∈ S(A), U(p) ≥ α + 1 if and only if for each cardinal number ë there is a set B ⊇ A and there are at least ë many types q(x) ∈ S(B) extending p and such that U(q) ≥ α. For a finite tuple a we will use the notation SU(a/A) = SU(tp(a/A)) and U(a/A) = U(tp(a/A)). Remark 13.6. SU is a foundation rank, the foundation rank of finitary complete types over sets with the relation of being a forking extension. In general, if R is a binary relation, the foundation rank of R is the mapping r assigning to every element of the domain of R an ordinal number (or ∞) according to the following rules: 1. r(a) ≥ 0. 2. r(a) ≥ α + 1 if and only if r(b) ≥ α for some b such that aRb. 3. r(a) ≥ α if and only if r(a) ≥ â for all â < α if α is a limit number. By induction on α (and induction on â in the case α + 1) one easily sees that 4. if r(a) ≥ α and α ≥ â then r(a) ≥ â, and therefore if one defines 5. r(a) = ∞ in case r(a) ≥ α for all α, 6. r(a) = sup{α : r(a) ≥ α} otherwise, it is clear that r(a) = α if and only if r(a) ≥ α and r(a) 6≥ α + 1. Some properties of SU are better understood if we bear in mind that it is a foundation rank. The following will be helpful: 7. If aRb and r(a) < ∞, then r(a) > r(b). 8. If R is transitive, the rank r is connected : if r(a) = α < ∞ and â < α, then r(b) = â for some b such that aRb. 9. If there is a sequence (ai : i < ù) such that a = a0 and ai Rai+1 for all i < ù, then r(a) = ∞. 10. If there is an ordinal number α such that for all a, r(a) ≥ α implies r(a) = ∞ then: if r(a) = ∞, then there is a sequence (ai : i < ù) such that a = a0 and ai Rai+1 for all i < ù. Proof. 7 is clear since, by definition, if r(b) ≥ α and aRb then r(a) ≥ α+1. 8 can be proved by induction on α using 7. For 9, prove that for all i, r(ai ) ≥ α for any α by induction on α. For 10 note that the hypothesis implies that if r(a) = ∞ then r(b) = ∞ for some b such that aRb. a Remark 13.7. SU(p) = 0 if and only if p is algebraic if and only if U(p) = 0. Proposition 13.8. Let T be simple and let p(x) ⊆ q(x) be complete finitary types. 1. If q is a nonforking extension of p, then SU(p) = SU(q).

13. Supersimple theories

87

2. If SU(p) = SU(q) < ∞, then q is a nonforking extension of p. Proof. 1. Clearly SU(p) ≥ SU(q). We now prove by induction on α that SU(p) ≥ α implies SU(q) ≥ α. Consider the case SU(p) ≥ α + 1. Let p(x) ∈ S(A) and q(x) ∈ S(B). For some C ⊇ A there is a forking extension p 0 ∈ S(C ) of p such that SU(p 0 ) ≥ α. Changing C (and p 0 ) if necessary, we may assume that there is some b |= q such that b |= p 0 and C ^ | Ab B. Then b ^ | C B, and hence q 0 = tp(b/CB) is a nonforking extension of p0 . By induction hypothesis, SU(q 0 ) ≥ α. Since q 0 is a forking extension of q, SU(q) ≥ α +1. Point 2 is clear and corresponds to point 7 of Remark 13.6. a Proposition 13.9. If T is stable, then U = SU. Proof. By Corollary 8.7 in a stable theory a global type p ∈ S(C) forks over A if and only if it has an unbounded orbit in Aut(C/A). By induction on α we prove that SU(p) ≥ α if and only if U(p) ≥ α. Consider the case α + 1. Assume p ∈ S(A), SU(p) ≥ α + 1 and q ∈ S(B) is a forking extension of p with SU(q) ≥ α. A nonforking extension q ∈ S(C) of q has unboundedly many A-conjugates. Fix ë and choose a set C ⊇ B such that q  C has ë many A-conjugates. By Proposition 13.8 and by the induction hypothesis U(q  C ) ≥ α and then all its A-conjugates over C also have U-rank ≥ α. This means that U(p) ≥ α + 1. For the other direction, assume U(p) ≥ α + 1 and choose ë > Mlt(p), the number of nonforking extensions of p. There is a set B ⊇ A over which p has ë extensions of U-rank ≥ α. By choice of ë, one of them, say q ∈ S(B), is a forking extension. By induction hypothesis SU(q) ≥ α. Then SU(p) ≥ α + 1. a Lemma 13.10. Let T be simple. 1. There is some ordinal α such that SU(p) ≥ α implies SU(p) = ∞. 2. If SU(p) = ∞, there is a forking extension q of p such that SU(q) = ∞. Proof. 1. Assume for every ordinal α there is a complete type pα (x) ∈ S(Aα ) such that α ≤ SU(pα ) < ∞. Since there is a subset B ⊆ Aα such that |B| ≤ |T | and pα does not fork over B, by Proposition 13.8 we may assume that in fact |Aα | ≤ |T |. For each α there are boundedly many types p(x) ∈ S(Aα ) and therefore there is an ordinal âα such that SU(p) ≤ âα if p(x) ∈ S(Aα ) and SU(p) < ∞. Fix an enumeration aα of Aα . Clearly âα = âα 0 if tp(aα ) = tp(aα 0 ). Since there are only boundedly many types tp(aα ) of such tuples aα , we can obtain an upper bound of all SU(pα ), which is a contradiction. 2 follows from 1 as shown in points 9, 10 of Remark 13.6. a Proposition 13.11. If T is simple, the following are equivalent for any finitary p ∈ S(A). 1. SU(p) = ∞. 2. There is a forking chain of types (pn : n < ù) starting with p = p0 . 3. Some q ∈ S(B) extending p forks over AB0 for any finite subset B0 ⊆ B.

88

13. Supersimple theories

Proof. 2 ⇔ 3: as in the proof of Proposition 4.13. 1 ⇔ 2 follows from Lemma 13.10 and points 9, 10 of Remark 13.6. a Remark 13.12. If p(x) ∈ S(M ) is not definable, then U(p) = ∞. Proof. As explained in the proof of Proposition 7.9 for each cardinal ë there is a model N  M over which there are ë different strong heirs of p. Since again they are all nondefinable, this can be used to show that U(p) = ∞. a Proposition 13.13. 1. T is supersimple if and only if SU(p) < ∞ for all finitary p. 2. T is superstable if and only if U(p) < ∞ for all finitary p. Proof. 1 follows from Proposition 13.11 and Proposition 13.4. 2. If T is superstable, T is stable and by Proposition 13.9 SU = U. Since T is also supersimple, by 1 U(p) < ∞ for all p. For the other direction, it is enough to show that stability follows from the condition U(p) < ∞ for all p. If T is not stable then there is a nondefinable finitary type p(x) ∈ S(M ) over some model M . Then we apply Remark 13.12. a Remark 13.14. If SU(p) = α < ∞, then for any â < α there is some q ⊇ p such that SU(q) = â. Proof. By point 8 of Remark 13.6.

a

Notation 13.15. We will denote by α ⊕ â the natural sum of the ordinals α, â. Every ordinal number α can be written uniquely in Cantor normal form Pk αi as α = 0 > · · · > αk are ordinals and n0 , . . . , nk are i=0 ù ni where αP j natural numbers > 0. If â = i=0 ù âi mi is also in Cantor normal form, then Pl α ⊕ â = i=0 ù ãi ri where ã0 > · · · > ãl enumerates α0 , . . . , αk , â0 , . . . , âj and   if ãi = αp 6∈ {â0 , . . . , âj } np ri = mp if ãi = âp 6∈ {α0 , . . . , αk }   np + mq if ãi = αp = âq . This sum is the least operation F : On × On → On which is strictly increasing in both arguments. Clearly, for natural numbers n, m, n + m = n ⊕ m. Theorem 13.16 (Lascar inequalities). Let T be simple. If SU(ab/A) < ∞, then SU(a/Ab) + SU(b/A) ≤ SU(ab/A) ≤ SU(a/Ab) ⊕ SU(b/A). Proof. It is easy to see by induction on α that if SU(a/A) ≥ α, then SU(ab/A) ≥ α. Hence SU(ab/A) ≥ SU(a/A). From SU(ab/A) < ∞ it follows then that SU(a/A) < ∞ and SU(b/A) < ∞. Then we can freely use Proposition 13.8. To check the inequality SU(ab/A) ≤ SU(a/Ab) ⊕ SU(b/A),

13. Supersimple theories

89

we prove by induction on α that if SU(ab/A) ≥ α, then SU(a/Ab) ⊕ SU(b/A) ≥ α. This is clear for α = 0 and for limit α. Let us consider the case α + 1. Assume SU(ab/A) ≥ α + 1. For some B ⊇ A we have SU(ab/B) ≥ α and ab ^ 6 | A B. Since ab ^ 6 | A B, either b ^ 6 | A B or a ^ 6 | Ab B. Therefore SU(b/A) > SU(b/B) or SU(a/Ab) > SU(a/Bb). By monotonicity of natural addition of ordinal numbers, SU(a/Ab) ⊕ SU(b/A) > SU(a/Bb) ⊕ SU(b/B). By the induction hypothesis SU(a/Bb) ⊕ SU(b/B) ≥ α. Hence SU(a/Ab) ⊕ SU(b/A) ≥ α + 1. To check the inequality SU(a/Ab) + SU(b/A) ≤ SU(ab/A), we show by induction on α that SU(ab/A) ≥ SU(a/Ab) + α if SU(b/A) ≥ α. The cases α = 0 and α limit are straightforward. For the case α + 1, assume SU(b/A) ≥ α +1. Then for some B ⊇ A, SU(b/B) ≥ α and b ^ 6 | A B. We may assume that B ^ | Ab a. By induction hypothesis SU(ab/B) ≥ SU(a/Bb) + α. Since b ^ 6 | A B, also ab ^ 6 | A B and then SU(ab/A) > SU(ab/B). Since a^ | Ab B we have SU(a/Ab) = SU(a/Bb). Therefore SU(ab/A) > SU(ab/ B) ≥ SU(a/Bb) + α = SU(a/Ab) + α. We then conclude SU(ab/A) ≥ SU(a/Ab) + α + 1. a Corollary 13.17. If T is simple and SU(ab/A) < ù, then SU(ab/A) = SU(a/Ab) + SU(b/A). Proof. As remarked above, for natural numbers n, m, n + m = n ⊕ m. a Proposition 13.18. Let T be simple. 1. If a ∈ acl(Ab), then SU(ab/A) = SU(b/A). 2. If acl(aA) = acl(bA) then SU(a/A) = SU(b/A). Proof. 1. Clearly SU(ab/A) ≥ S(b/A). Moreover it is easy to check by induction on α that SU(ab/A) ≥ α implies SU(b/A) ≥ α. 2 follows from 1. a Definition 13.19. An abstract rank is a mapping R assigning an ordinal number or ∞ to all finitary complete types over sets and satisfying the following conditions: 1. If f ∈ Aut(C), then R(p) = R(p f ). 2. If p ⊆ q, then R(p) ≥ R(q). 3. If p ∈ S(A) and A ⊆ B, then there is some extension q ∈ S(B) of p such that R(p) = R(q). 4. Let p ∈ S(A) be such that R(p) < ∞. There is a cardinal κ such that for each B ⊇ A, p has at most κ extensions q ∈ S(B) such that R(p) = R(q). Remark 13.20. Let R be an abstract rank. If p ∈ S(M ) is not definable, then R(p) = ∞.

90

13. Supersimple theories

Proof. Choose α minimal for which there is some nondefinable p ∈ S(M ) over some model M with R(p) = α. Let κ be the cardinal given by condition 4 in the definition of rank. As shown in the proof of Proposition 7.9 there is a model N  M over which there are κ + different strong heirs of p. All are nondefinable and one of them must have rank < α, a contradiction. a Proposition 13.21. Let R be an abstract rank. 1. Let T be stable, p ⊆ q, and R(p) < ∞. Then R(p) = R(q) if and only if q is a nonforking extension of p. 2. If R(p) < ∞ for every complete type p, then T is superstable. Proof. 1. Let p ∈ S(A), A ⊆ B, and p ⊆ q ∈ S(B). We assume T is stable and R(p) < ∞. Fix κ, a bound for the extensions of p of rank R(p). We can find a model M ⊇ A such that all nonforking extensions of p over M are A-conjugate in M and such that each forking extension of p over M has more than κ A–conjugates in M . There is an extension q 0 ∈ S(M ) of q with R(q) = R(q 0 ). Now, if q forks over A then also q 0 forks and therefore q 0 has more than κ A-conjugates. By definition of rank R(p) > R(q 0 ). Now assume R(p) > R(q) and q does not fork over A. Let q 0 ∈ S(M ) be a nonforking extension of q and choose r ∈ S(M ), an extension of p of rank R(p) = R(r). As shown above, r does not fork over A. By choice of M , q 0 and r are A-conjugate. Hence R(p) = R(r) = R(q 0 ) = R(q). 2. It suffices to show the stability of T , since we then can use point 1 to easily verify that T is supersimple. If T is unstable then some finitary type p ∈ S(M ) is nondefinable. By Remark 13.20 R(p) = ∞. a Proposition 13.22. In a stable theory U is an abstract rank and it is minimal, that is, U(p) ≤ R(p) for any other abstract rank R. Proof. If T is stable, then U = SU and by Proposition 13.8 whenever p ⊆ q and U(p) < ∞, q is a nonforking extension of p if and only if U(p) = U(q). Since in a stable theory a type has only a bounded number of nonforking extensions, the requirements in the definition of abstract rank are fulfilled. Minimality is easily checked showing by induction on α that if R is a rank and U(p) ≥ α, then R(p) ≥ α. a Corollary 13.23. T is superstable if and only if there is an abstract rank R such that R(p) < ∞ for all p. Proof. If T is superstable, then U is an abstract rank and U(p) < ∞ for all p. The rest follows from Proposition 13.21. a Proposition 13.24. Let T be stable and let p(x) ∈ S(A) be finitary. 1. If U(p) < ∞, then for any B ⊇ A there are at most 2|T | + |B| extensions q(x) ∈ S(B) of p. 2. If U(p) = ∞ then for any cardinal ë ≥ |T | + |A| there is a set B ⊇ A such that |B| ≤ ë and p has at least ëù extensions q(x) ∈ S(B).

13. Supersimple theories

91

Proof. If T is stable, then U = SU. By Proposition 13.11, if U(p) < ∞ then any complete type q over B ⊇ A extending p does not fork over AB0 for some finite B0 ⊆ B. Since there are only 2|T | extensions of p to a complete type q(x) ∈ S(AB0 ) for B0 finite, and since each such type q has at most 2|T | nonforking extensions over B, it is easy to check that 2|T | + |B| is a correct upper bound for the number of extensions of p over B. On the other hand, if U(p) = ∞ by Lemma 13.10 p has a forking extension q of U-rank ∞. Let q be a global nonforking extension of q. Then q forks over A and therefore it has an unbounded orbit in Aut(C/A). Note that every complete type between p and q has U-rank ∞. Fix a set A1 ⊇ A such that |A1 | ≤ ë and for which there are different types ri (x) ∈ S(A1 ) for i < ë which can be extended to A-conjugates of q. Note that U(ri ) = ∞. Iterating this procedure we obtain a chain of sets (An : n < ù) of cardinality |An | ≤ ë and a tree of types 0 (ps : s ∈ ë<ù ) such that ps ∈ S(An ) if s ∈ ën , p∅ = S p, ps ⊆ ps 0 if s ù⊆ s , 0 ps 6= ps 0 if s 6= s and U(ps ) = ∞. If we put pf = s⊆f ps for f ∈ ë , we obtain aS family (pf : f ∈ ëù ) of ëù many complete extensions of p over the set B = n<ù An of cardinality |B| ≤ ë. a Theorem 13.25. The following are equivalent: 1. T is superstable. 2. For all A, for all n < ù, |Sn (A)| ≤ |A| + 2|T | . 3. For all ë ≥ 2|T | , T is ë-stable. 4. There is some cardinal ì such that for all ë ≥ ì, T is ë-stable. Proof. 1 ⇒ 2. There are only 2|T | finitary types over ∅, and by Proposition 13.24 and Proposition 13.13 each finitary p(x) ∈ S(∅) has at most 2|T | + |A| complete extensions over A. It is clear that 2 ⇒ 3 and 3 ⇒ 4. 4 ⇒ 1. If T is not superstable, then, by Proposition 13.13, there is some p(x) ∈ S(A) such that U(p) = ∞. Choose ë ≥ ì + |T | + |A| such that ëù > ë. By Proposition 13.24 there is a set B ⊇ A of cardinality ≤ ë such that p has at least ëù complete extensions over B. Clearly T is not ë-stable. a

Chapter 14

MORE RANKS

Definition 14.1. D-rank is defined for formulas ϕ(x) ∈ L(C) (in finitely many variables x) as follows: 1. D(ϕ(x)) ≥ 0 if and only if ϕ(x) is consistent. 2. D(ϕ(x)) ≥ α + 1 if and only if for some ø(x, y) ∈ L for all cardinal numbers ë there is an infinite sequence (ai : i < ë) such that (a) {ø(x, ai ) : i < ë} is k-inconsistent for some k < ù, (b) |= ø(x, ai ) → ϕ(x) for each i < ë, and (c) D(ø(x, ai )) ≥ α for each i < ë. 3. D(ϕ(x)) ≥ â if and only if D(ϕ(x)) ≥ â for all â < α for limit α. If ϕ(x) is inconsistent, D(ϕ) = −1 and otherwise it is the supremum of all α such that D(ϕ) ≥ α. The definition is extended to arbitrary sets of formulas ð(x) by D(ð(x)) = min{D(ϕ) : ϕ is a finite conjunction of formulas in ð(x)}. Remark 14.2. If ϕ(x) ∈ L(A), then D(ϕ(x)) ≥ α + 1 if and only if |= ø(x) → ϕ(x) and D(ø(x)) ≥ α for some ø(x) ∈ L(C) which divides over A. Proposition 14.3. 1. There is an ordinal α such that for all ϕ(x) ∈ L(C), if D(ϕ) ≥ α, then D(ϕ) = ∞. 2. If ϕ(x) ∈ L(A) and D(ϕ(x)) = ∞, then D(ø(x)) = ∞ for some ø(x) such that |= ø(x) → ϕ(x) and ø(x) divides over A. 3. D(ϕ(x)) = ∞ if and only if there is a sequence (ϕi (x) : i < ù) of consistent formulas ϕi (x)S∈ L(Ai ) such that ϕ = ϕ0 , |= ϕi+1 (x) → ϕi (x) and ϕi+1 (x) divides over j≤i Aj . 4. T is supersimple if and only if D(ϕ) < ∞ for all ϕ. Proof. 1 is easy, as in Lemma 13.10, 2 follows from 1, and 3 follows from 2. Lastly, 4 follows from 3 and Proposition 13.4. a Lemma 14.4. 1. If ð1 (x) ` ð2 (x), then D(ð1 ) ≤ D(ð2 ). 2. D(ð) = 0 if and only if ð is algebraic. 3. D(ϕ ∨ ø) = max{D(ϕ), D(ø)}. 4. If ð(x) is a partial type over A, there is some p(x) ∈ S(A) such that ð ⊆ p and D(ð) = D(p). 93

94

14. More ranks

5. If ð(x) is a partial type, there is some finite conjunction ϕ(x) of formulas of ð(x) such that D(ð) = D(ϕ). Proof. 4 follows from 3. As for 3, it is clear that D(ϕ), D(ø) ≤ D(ϕ ∨ ø). Then it suffices to show that if D(ϕ ∨ ø) ≥ α, then D(ϕ) ≥ α or D(ø) ≥ α, and this can be shown by induction on α. Consider the case α + 1. Assume D(ϕ ∨ ø) ≥ α + 1. For some è and A, |= è → (ϕ ∨ ø), (ϕ ∨ ø) ∈ L(A), è divides over A, and D(è) ≥ α. Note that |= è ↔ (è ∧ ϕ) ∨ (è ∧ ø) and hence the induction hypothesis gives D(è ∧ ϕ) ≥ α or D(è ∧ ø) ≥ α. We conclude D(ϕ) ≥ α + 1 or D(ø) ≥ α + 1. a Remark 14.5. If T is simple, then SU ≤ D. Proof. By induction on α it is easy to check that SU(p) ≥ α implies D(p) ≥ α. a Remark 14.6. 1. Let T be simple. If p(x) ∈ S(B) forks over A ⊆ B, and D(p) < ∞, then D(p) < D(p  A). 2. There is an example of a supersimple theory where there is a type p(x) ∈ S(A) of D(p) = 2 having a nonforking extension q(x) ∈ S(B) such that D(q) = 1. Proof. 1. Let α = D(p), choose ϕ(x) ∈ p with D(ϕ) = D(p  A) and choose ø(x) ∈ p which forks over A. Then ø 0 (x) = ø(x) ∧ ϕ(x) ∈ p and hence D(ø 0 (x)) ≥ α. Since ø 0 (x) forks over A and implies ϕ(x), D(p  A) = D(ϕ(x)) ≥ α + 1. For 2 see Example 5.1.15 in [41]. a Definition 14.7. The continuous rank RC (also denoted R∞ ) is defined for all sets of formulas (in finitely many variables) as follows: 1. RC(ð(x)) ≥ 0 if and only if ð(x) is consistent. 2. RC(ð(x)) ≥ α + 1 if and only if for any conjunction ϕ(x) of formulas in ð(x) for any cardinal ë there is a sequence (ði (x) : i < ë) of partial types ði (x) 3 ϕ(x) such that RC(ði ) ≥ α and ði ∪ ðj is inconsistent for all i < j < ë. 3. RC(ð(x)) ≥ â if and only if RC(ð(x)) ≥ α for all α < â if â is a limit number. For a formula ϕ(x) we set RC(ϕ) = RC({ϕ}). Lemma 14.8. 1. If ð(x) ` ð0 (x), then RC(ð) ≤ RC(ð0 ). 2. RC(ð) = 0 if and only if ð is algebraic. 3. If ð(x) is a partial type over A, RC(ð) = min{RC(ϕ) : ϕ is a finite conjunction of formulas in ð} and therefore there is a finite conjunction ϕ(x) of formulas in ð(x) such that RC(ð) = RC(ϕ). 4. RC(ð ∪ {(ϕ ∨ ø)}) = max{RC(ð ∪ {ϕ}), RC(ð ∪ {ø})}. 5. If ð(x) is a partial type over A, there is some p(x) ∈ S(A) such that ð ⊆ p and RC(ð) = RC(p).

14. More ranks

95

Proof. 1. It is an induction on α: if RC(ð) ≥ α, then RC(ð0 ) ≥ α. In the case α + 1, given ϕ, a conjunction of formulas in ð0 , and given a cardinal ë, we first find ø, a conjunction of formulas in ð such that ø ` ϕ; next we use the hypothesis RC(ð) ≥ α + 1 to find a sequence (ði (x) : i < ë) of pairwise incompatible types ði 3 ø with RC(ði ) ≥ α, and then we set ði0 = ði ∪ {ϕ}. Since ði ` ði0 , by induction hypothesis RC(ði0 ) ≥ α. Hence (ði0 : i < ë) witnesses that RC(ð0 ) ≥ α + 1. For 3, choose ϕ, a conjunction of formulas in ð of minimal RC-rank, and show by induction on α that RC(ϕ) ≥ α implies RC(ð) ≥ α. 4. By 1 it is clear that RC(ð ∪ {ϕ ∨ ø}) ≥ max{RC(ð ∪ {ϕ}), RC(ð ∪ {ø})} ≥ α. Hence we only have to show that if RC(ð ∪ {ϕ ∨ ø}) ≥ α, then max{RC(ð ∪ {ϕ}), RC(ð ∪ {ø})} ≥ α, and this can be done by induction on α. As usual, we consider only the case α + 1. Assume RC(ð ∪ {ϕ}) 6≥ α + 1 and RC(ð ∪ {ø}) 6≥ α + 1. Hence we have ä1 , ä2 , conjunctions of formulas in ð, and ë1 , ë2 , cardinal numbers, such that there is no sequence (ði : i < ë1 ) of pairwise incompatible types ði 3 (ä1 ∧ ϕ) with RC(ði ) ≥ α and there is no sequence (ði : i < ë2 ) of pairwise incompatible types ði 3 (ä2 ∧ ø) with RC(ði ) ≥ α. Let ä = (ä1 ∧ ä2 ) and let ë = max{ë1 , ë2 }. There is a sequence (ði : i < ë) of pairwise incompatible types ði 3 (ä ∧(ϕ ∨ø)) with RC(ði ) ≥ α. Note that ði ≡ ði ∪ {ä} ∪ {ϕ ∨ ø} and then, by 1, RC(ði ∪ {ä} ∪ {ϕ ∨ ø}) ≥ α and by induction hypothesis either RC(ði ∪ {ä} ∪ {ϕ}) ≥ α or RC(ði ∪ {ä} ∪ {ø}) ≥ α. Again by 1, either RC(ði ∪ {ä1 } ∪ {ϕ}) ≥ α or RC(ði ∪ {ä2 } ∪ {ø}) ≥ α. One of these two possibilities takes place ë times, contradicting the choice of ë1 and ë2 . 5 follows from 4 as in other similar situations. a Remark 14.9. RC(ð(x)) ≥ α + 1 if and only if for each ϕ(x), conjunction of formulas of ð, for each cardinal ë there is a set A and there is a family (pi (x) : i < ë) of different types pi (x) ∈ S(A) such that RC(pi ) ≥ α for all i < ë. Proof. By point 5 of Lemma 14.8.

a

Proposition 14.10. If T is stable, then D = RC. Proof. It is enough to check it for formulas and then it is clear: after Corollary 8.7, for stable T and ϕ(x) ∈ L(A), RC(ϕ(x)) ≥ α + 1 if and only if there is some ø(x) such that |= ø(x) → ϕ(x), ø(x) forks over A, and RC(ϕ) ≥ α. a Proposition 14.11. T is superstable if and only if RC(ϕ) < ∞ for every ϕ.

96

14. More ranks

Proof. One direction follows from Proposition 14.10 and point 4 of Proposition 14.3. For the other direction note that U(p) ≤ RC(p) for any complete type p, and then apply Proposition 13.13. a Definition 14.12. An abstract rank R is a continuous rank if for every α, for every A, {p(x) ∈ S(A) : R(p) < α} is an open subset of S(A). Proposition 14.13. If T is stable, RC is the smallest continuous rank in T . Proof. By definition and by Lemma 14.8 it is clear that RC always satisfies conditions 1–3 of abstract rank. For condition 4 we need to assume T is stable. By Proposition 14.10 RC = D. If p(x) ∈ S(A), RC(p) = α < ∞, and q is a forking extension of p of the same rank RC(q) = α, then q contains a formula ϕ(x) which forks over A. We can assume that RC(ϕ) = α and that ϕ implies some ø(x) ∈ p of rank RC(ø) = α. But then D(ø) ≥ α + 1, which is a contradiction. Therefore, all extensions q of p with RC(q) = α are nonforking extensions and by stability its number is bounded by the multiplicity of p, which is ≤ 2|T | . It follows that RC is an abstract rank. Point 3 of Lemma 14.8 implies that RC is continuous. If R is another continuous rank, then by induction on α one sees that if RC(p) ≥ α then R(p) ≥ α. Consider the case α + 1. Let p(x) ∈ S(A) be such that RC(p) ≥ α + 1. We will show that for any ϕ ∈ p there is some q ∈ S(A) such that ϕ ∈ q and R(q) ≥ α + 1. Continuity of R will then imply R(p) ≥ α + 1. Now, by Remark 14.9 for each cardinal ë there is some B such that there are at least ë types q(x) ∈ S(B) such that ϕ(x) ∈ q and RC(q) ≥ α. We may assume that always A ⊆ B. Since there are only 2|T |+|A| types over A, for some r(x) ∈ S(A) such that ϕ ∈ r and for each cardinal ë there is some B such that there are at least ë types q(x) ∈ S(B) such that r ⊆ q and RC(q) ≥ α. By induction hypothesis R(q) ≥ α for all such q. By condition 4 in the definition of abstract rank R(r) ≥ α + 1. a Definition 14.14. The Morley rank of a global type p ∈ Sn (C), RM(p), is its Cantor–Bendixson rank in the space Sn (C). The Morley rank of a partial type ð(x), RM(ð), (where x is a n-tuple of variables) is the Cantor–Bendixson rank of the closed set {p ∈ Sn (C) : ð ⊆ ð} and its Morley degree, DM(ð), is the Cantor–Bendixson degree of this closed set. By compactness, DM(ð) is finite if RM(ð) < ∞. It is clear that RM(ð) = max{RM(p) : ð ⊆ p}. For a formula ϕ we set RM(ϕ) = RM({ϕ} and DM(ϕ) = DM({ϕ}). Remark 14.15. 1. RM(ϕ(x)) ≥ 0 if and only if ϕ(x) is consistent. 2. RM(ϕ(x)) ≥ α + 1 if and only if there is a sequence (ϕi (x) : i < ù) such that |= ϕi (x) → ϕ(x), RM(ϕi ) ≥ α, and ϕi (x) ∧ ϕj (x) is inconsistent for all i 6= j. 3. RM(ϕ) ≥ α if and only if RM(ϕ) ≥ â for all â < α if α is a limit number.

14. More ranks

97

Proof. By Proposition 1.17. a Remark 14.16. RM(ϕ(x)) ≥ α + 1 if and only if there for each n < ù there is a sequence (ϕi (x) : i < n) such that |= ϕi (x) → ϕ(x), RM(ϕi ) ≥ α, and ϕi (x) ∧ ϕj (x) is inconsistent for all i 6= j. Hence the degree DM(ϕ(x)) can be defined (in case RM(ϕ) = α < ∞) as the maximal n for which there is a sequence (ϕi (x) : i < n) such that |= ϕi (x) → ϕ(x), RM(ϕi ) ≥ α, and ϕi (x) ∧ ϕj (x) is inconsistent for all i 6= j. Proof. By point 4 of Proposition 1.17. a Proposition 14.17. For any partial type ð, 1. RM(ð) = min{RM(ϕ) : ϕ is a conjunction of formulas in ð}, 2. DM(ð) = min{DM(ϕ) : ϕ is a conjunction of formulas in ð and RM(ϕ) = RM(ð)}. Proof. By Proposition 1.16. a Remark 14.18. Morley rank can be computed in any ù-saturated model M containing the parameters of the type as the Cantor–Bendixson rank in S(M ) of the closed set determined by the type. Proof. It is enough to check it for formulas and in this case we can use Remark 14.15. The parameters needed in the sequence (ϕi (x) : i < ù) to check that RM(ϕ(x)) ≥ α + 1 build a countable sequence and its type over the parameters of ϕ can be realized in M . a Proposition 14.19. Morley rank is a continuous rank. Proof. All conditions in the definition of an abstract rank are easily seen to be satisfied by Morley rank. The bound for the number of extensions with the same rank of a type p(x) is DM(p). Continuity follows from Proposition 14.17. a Corollary 14.20. In a stable theory, U ≤ RC ≤ RM. Proof. By propositions 14.19, 13.22, and 14.13. a Definition 14.21. T is totally transcendental if and only if RM(ϕ) < ∞ for all ϕ. Theorem 14.22. 1. If T is ë-stable for some ë < 2ù , then T is totally transcendental. 2. Any totally transcendental theory is ë-stable for all ë ≥ |T |. Proof. 1. Assume RM(ϕ) = ∞. By Remark 14.18, we can work in S(M ) for some model M such that ϕ(x) ∈ L(M ) and we can then apply Proposition 1.18, obtaining a tree of formulas (ϕs : s ∈ 2<ù ) such that ϕ∅ = ϕ, RM(ϕs ) = ∞, ϕs ≡ (ϕs a 0 ∨ ϕs a 1 ) and (ϕs a 0 ∧ ϕs a 1 ) is inconsistent. Every branch f ∈ 2ù gives rise to a type ðf = {ϕs : s ⊆ f} and this produces a set of 2ù incompatible partial types over a countable set of parameters, contradicting ë-stability of T if ë < 2ù .

98

14. More ranks

2. Let ë ≥ |T | and let |A| ≤ ë. For each p(x) ∈ S(A) choose some ϕp (x) ∈ S(A) such that RM(p) = RM(ϕp ) and DM(p) = DM(ϕp ). Since T is totally transcendental, for any ø(x) ∈ L(A), ø ∈ p if and only if RM(ϕp ∧ ø) = RM(ϕp ) and DM(ϕp ∧ ø) = DM(ϕp ). It follows that p 6= q implies ϕp 6= ϕq . Hence the number |T | + |A| of formulas ϕ(x) ∈ L(A) is an upper bound for |S(A)|. a Corollary 14.23. Totally transcendental theories, and in particular ù-stable theories, are superstable. Proof. By theorems 14.22 and 13.25. a Definition 14.24. T is small if for all n < ù, |Sn (∅)| ≤ ù. Remark 14.25. If T is small, then there is a countable L0 ⊆ L such that for every ϕ(x) ∈ L there is some ϕ 0 (x) ∈ L0 such that in T , ϕ(x) ≡ ϕ 0 (x). Proof. Let Sn (∅) = {pi (x) : i < ù}. Choose ϕi,j (x) ∈ pi such that ¬ϕi,j(x) ∈ pj whenever pi 6= pj . If a, b satisfy the same formulas of {ϕi,j : i, j < ù} then they have the same type over ∅ and therefore |= ϕ(a) if and only if |= ϕ(b) for every ϕ(x) ∈ L. Hence, every such ϕ(x) is a boolean combination of the formulas ϕi,j (x). a Remark 14.26. The following are equivalent: 1. T is small. 2. For all n < ù, for all finite A, |Sn (A)| ≤ ù. 3. For all finite A, |S1 (A)| ≤ ù. 4. T has a saturated countable model. Proof. 1 ⇒ 2 can be justified by a standard counting types argument. 2 ⇒ 3 is clear. S For 3 ⇒ 4, the countable saturated model can be constructed as a union n∈ù An of countable sets An such that each complete 1-type over a finite subset of An is realized in An+1 . 4 ⇒ 1 is clear since all p(x) ∈ Sn (∅) can be realized in the countable saturated model. a Remark 14.27. A theory T is small if and only if the two following conditions hold : 1. For all n < ù, for all finite A, the space Sn (A) is scattered, that is, every type in Sn (A) has ordinal Cantor–Bendixson rank. 2. There is a countable L0 ⊆ L such that for every ϕ(x) ∈ L there is some ϕ 0 (x) ∈ L0 such that in T , ϕ(x) ≡ ϕ 0 (x). Proof. By Remark 14.25 and Corollary 1.19 any small theory satisfies 1 and 2. For the opposite direction, assume 1 and 2 hold in T . By 2, the space Sn (∅) has a countable basis of open sets. By 1 and Proposition 1.20, Sn (∅) is countable. a Remark 14.28. 1. ù-categorical theories are small. 2. ù-stable theories are small. Proof. Clear. a

14. More ranks

99

Corollary 14.29. T is ù-stable if and only if T is small, superstable, and every complete type has finite multiplicity. Proof. Let T be ù-stable. By Remark 14.28 T is small and by Corollary 14.23 T is superstable. By Theorem 14.22 T is totally transcendental and therefore the multiplicity of a type is its Morley degree. The other direction is just a counting types argument as in the proof of Theorem 13.25. a Corollary 14.30. Superstable ù-categorical theories are ù-stable. Proof. By Corollary 14.29, since by ù-categoricity for each finite A there are only finitely many complete n-types over A. Moreover by ù-categoricity if A is finite there is a finest finite A-definable equivalence on relation on n-tuples and, together with the Finite Equivalence Relation Theorem 9.30, this implies that all multiplicities of complete types over A are finite. a Definition 14.31. An abstract rank R is cantorian if and only if any type p(x) ∈ S(A) has rank R(p) ≥ α + 1 in the case that p is an accumulation point of {q(x) ∈ S(A) : R(q) ≥ α}. Proposition 14.32. Let p(x) ∈ S(A). Then RM(p) ≥ α + 1 if and only if for some B ⊇ A some extension q(x) ∈ S(B) of p is an accumulation point of {r(x) ∈ S(B) : RM(r) ≥ α}. Proof. Let RM(p) ≥ α + 1 and choose an ù-saturated model M ⊇ A and let q(x) ∈ S(M ) be an extension of p of Morley rank ≥ α + 1. By Remark 14.18 q has Cantor–Bendixson rank ≥ α + 1 in S(M ) and therefore it is an accumulation point of types r(x) ∈ S(M ) of Cantor–Bendixson rank ≥ α. Again by Remark 14.18, these types r have Morley rank ≥ α. For the other direction it is enough to prove that RM(q) ≥ α + 1, in other words, that RM is cantorian. For this it is enough to show that each ϕ ∈ q is contained in some q ∈ S(C) of Cantor–Bendixson rank ≥ α + 1, that is, ϕ is contained in infinitely many q ∈ S(C) of Cantor–Bendixson rank ≥ α. We know that each such ϕ is contained in infinitely many types r(x) ∈ S(B) of Morley rank ≥ α. But we can choose for each such r(x) ∈ S(B) an extension q(x) ∈ S(C) of Cantor–Bendixson rank ≥ α. a Proposition 14.33. RM is the smallest cantorian rank. Proof. By propositions 14.19 and 14.32, RM is a cantorian rank. Let R be another cantorian rank. We prove by induction on α that RM(p) ≥ α implies R(p) ≥ α. Consider the case α + 1. Assume p(x) ∈ S(A) and RM(p) ≥ α + 1. By Proposition 14.32 for some B ⊇ A, some q(x) ∈ S(B) extending p is an accumulation point of {r(x) ∈ S(B) : RM(r) ≥ α}. By induction hypothesis, this set is contained in {r(x) ∈ S(B) : R(r) ≥ α} and hence q is an accumulation point of this set. Since R is cantorian, R(q) ≥ α+1 and therefore R(p) ≥ α + 1. a Theorem 14.34. If T is superstable and ù-categorical, then U is cantorian and therefore U = RC = RM.

100

14. More ranks

Proof. Let p(x) ∈ S(A) be an accumulation point of {q(x) ∈ S(A) : U(q) ≥ α}. By corollaries 14.29 and 14.23, every type has finite multiplicity and hence we can find a finite subset A0 ⊆ A such that p does not fork over A0 and p0 = p  A0 has p as its only nonforking extension over A. By ùcategoricity, the type p0 (x) is isolated by some ϕ0 (x) ∈ p0 . By assumption, there is some q(x) ∈ S(A) such that ϕ0 (x) ∈ q, U(q) ≥ α and p 6= q. It follows that q forks over A0 . Hence U(p) = U(p0 ) ≥ U(q) + 1 ≥ α + 1. The rest follows from Proposition 14.33 and Corollary 14.20. a

Chapter 15

HYPERIMAGINARIES

Definition 15.1. For any set A, an A-hyperimaginary is an equivalence class [a]E of a possibly infinite tuple a under a type-definable over A equivalence relation E. We will say that it is a hyperimaginary of sort E. In order to simplify notation we set aE = [a]E and we often identify the equivalence relation E with the partial type over A which defines E. Without loss of generality we can always assume that the type E(x, y) is closed under conjunction and that all formulas ϕ(x, y) ∈ E(x, y) are symmetric, that is ` ϕ(x, y) → ϕ(y, x). Clearly, A-imaginaries are A-hyperimaginaries. A hyperimaginary is a ∅hyperimaginary. Thus, real elements and imaginaries are a special case of hyperimaginaries. We sometimes use Cheq for the class of all hyperimaginaries. If a is a sequence a = (ai : i < α) of elements ai for some ordinal α, we say that α is the length of the hyperimaginary aE . Finitary hyperimaginaries are hyperimaginaries of finite length. Countable hyperimaginaries are hyperimaginaries of countable length. Definition 15.2. An automorphism f ∈ Aut(C) fixes a hyperimaginary aE if f(aE ) = aE , that is, if |= E(a, f(a)). Let A be a set of hyperimaginaries. A hyperimaginary e is definable over A if e is fixed by all automorphisms fixing A pointwise. The definable closure of A, dclheq (A), is the class of all hyperimaginaries definable over A. Since a hyperimaginary can have any length, dclheq (A) is a proper class. As usual, if a is a sequence of hyperimaginaries, dclheq (a) is defined as dclheq (A) where A is the set enumerated in a. As in the case of real elements, we extend the definition to arbitrary classes A by letting dclheq (A) be the union of dclheq (B) for all subsets B ⊆ A. Notice that if A is a class of imaginaries then dcleq (A) = dclheq (A)∩Ceq . We say that the sequences of hyperimaginaries a and b are equivalent if dclheq (a) = dclheq (b). This means that a, b are interdefinable, which is clearly equivalent to Aut(C/a) = Aut(C/b). In this situation we write a ∼ b. Lemma 15.3. Any sequence of hyperimaginaries is equivalent to a hyperimaginary. Proof. Let a = ([ai ]Ei : i ∈ I ) be a sequence of hyperimaginaries, where Ei is an equivalence relation on Ji -sequences and ai = (a(i,j) : j ∈ Ji ). 101

102

15. Hyperimaginaries S

Put K = i∈I {i} × Ji and consider the equivalence relation E defined for x = (x(i,j) : (i, j) ∈ K ) and y = (y(i,j) : (i, j) ∈ K ) by ^ E(x, y) ↔ Ei ((x(i,j) : j ∈ Ji ), (y(i,j) : j ∈ Ji )). i∈I

Clearly e = [(a(i,j) : (i, j) ∈ K )]E is a hyperimaginary and a ∼ e. a Lemma 15.4. Any hyperimaginary is equivalent to a sequence of countable hyperimaginaries. Proof. Recall that we may assume that the type E(x, y) defining the equivalence relation E is closed under conjunction and all its formulas are symmetric. It will be enough to find for each ϕ(x, y) ∈ E(x, y) a countable partial type Eϕ (x, y) ⊆ E(x, y) containing ϕ(x, y) which defines an equivalence relation. Given ϕ(x, y) ∈ E(x, y) we set Eϕ = {ϕn : n ∈ ù}, where ϕ0 = ϕ and ϕn+1 (x, y) ∈ E(x, y) satisfies ` ϕn+1 (x, y) ∧ ϕn+1 (y, z) → ϕn (x, z). The existence of such a ϕn+1 follows by compactness from the fact that E(x, y) ∪ E(y, z) ` ϕn (x, z). a Lemma 15.5. Let ð(x) be a partial type over A. If E is an equivalence relation on realizations of ð and it is type-definable over A, then there exists an equivalence relation F defined for all sequences of the length of x which is type-definable over A and agrees with E in ð(C). Proof. Set F (x, y) ⇔ (ð(x) ∧ ð(y) ∧ E(x, y)) ∨ x = y. a Proposition 15.6. Let e be a hyperimaginary and let b be a real tuple. If e ∈ dclheq (b), then e ∼ bE for some 0-type-definable equivalence relation E. Proof. Let e = aF . Since aF is type-definable over a and it is b-invariant, it is type-definable over b and there is a partial type ð(x, y) over ∅ such that ð(x, b) defines aF . Let p(y) = tp(b). If b 0 |= p then ð(x, b 0 ) defines an F class, and hence either defines e or a class disjoint with it. Thus ∃x(ð(x, y) ∧ ð(x, z)) defines an equivalence relation G(y, z) in p(C). By Lemma 15.5 there is an equivalence relation E which is type-definable over ∅ and agrees with G in p(C). It is easy to see that e ∼ bE . a Corollary 15.7. If e ∈ dclheq (A) for some set A of cardinality ≤ κ then e is equivalent to a hyperimaginary of length ≤ κ. Proof. It follows from Proposition 15.6. a Definition 15.8. The algebraic closure of A, a set of hyperimaginaries, is the class aclheq (A) consisting of all hyperimaginaries having finite orbit under the group of all automorphisms fixing A pointwise, that is aclheq (A) = {b ∈ Cheq : |{f(b) : f ∈ Aut(C/A)}| < ù}. The bounded closure of A is the class bdd(A) consisting of all hyperimaginaries having a bounded orbit under the group of all automorphisms fixing A

15. Hyperimaginaries

103

pointwise, that is bdd(A) = {b ∈ Cheq : |{f(b) : f ∈ Aut(C/A)}| < |C|}. As usual, if a enumerates A we put aclheq (a) = aclheq (A) and bdd(a) = bdd(A). Moreover we extend the definitions aclheq (A) and bdd(A) to arbitrary classes A of hyperimaginaries as customary: we take the union of all aclheq (B) (respectively bdd(B)) where B ranges over subsets of A. A hyperimaginary b is A-bounded if b ∈ bdd(A) and it is bounded if it is ∅-bounded. The hyperimaginaries a, b are interbounded if bdd(a) = bdd(b). Remark 15.9. Ceq ∩ bdd(A) = Ceq ∩ aclheq (A) = acleq (A) for any class of imaginaries A. Proof. By compactness, if an imaginary has infinitely many conjugates over a set A it has unboundedly many. a Definition 15.10. We now define the type tp(aE /bF ) of a hyperimaginary aE over some hyperimaginary bF . For each formula ϕ(x, y) ∈ L let Φϕ (x, y) be the set of formulas ∃x 0 y 0 (E(x, x 0 ) ∧ F (y, y 0 ) ∧ ϕ(x 0 , y 0 )). We define tp(aE /bF ) as the union of all partial types Φϕ (x, b) such that |= ϕ(a 0 , b 0 ) for some a 0 , b 0 such that E(a, a 0 ) and F (b, b 0 ). It is a partial type over b but the choice of another representative b 00 in the F -class of b gives an equivalent partial type over b 00 . As usual, for hyperimaginaries e, c, d we write e ≡c d for tp(e/c) = tp(d/c) (after choosing the same representative of c). Proposition 15.11. The following are equivalent: 1. aE ≡cF bE . 2. ac ≡ a 0 c 0 for some a 0 , c 0 such that E(a 0 , b) and F (c 0 , c). 3. a 0 c 0 ≡ b 0 c 00 for some a 0 , c 0 , b 0 , c 00 such that E(a 0 , a), F (c 0 , c), E(b 0 , b) and F (c 00 , c). 4. There is some f ∈ Aut(C/cF ) such that f(aE ) = bE . Proof. 4 ⇒ 1, 2 ⇒ 3 and 3 ⇒ 4 are clear. For 1 ⇒ 2, notice that if tp(aE /cF ) = tp(bE /cF ) and p(x, y) = tp(a, c), then ð(x, y) = E(x, b) ∪ F (y, c) ∪ p(x, y) is consistent. If |= ð(a 0 , c 0 ), then E(a 0 , b), F (c 0 , c) and ac ≡ a 0 c 0 .

a

Remark 15.12. Types can also be defined for sequences of hyperimaginaries in an analogous way. Assume a = ([ai ]Ei : i ∈ I ) and b = ([bj ]Fj : j ∈ J ) are sequences of hyperimaginaries. Let x = (xi : i ∈ I ) and y = (yj : j ∈ J ) where xi is of the length of ai and yj is of the length of bj . For each ϕ(x, y) ∈ L let x 0 = (xi0 : i ∈ I ) and y 0 = (yj0 : j ∈ J ) be new sequences of variables and

104

15. Hyperimaginaries

let Φϕ (x, y) be ∃(xi0 : i ∈ I )(yj0 : j ∈ J )(

^

i∈I

Ei (xi , xi0 ) ∧

^

Fj (yj , yj0 ) ∧ ϕ(x 0 , y 0 )).

j∈J

The type tp(a/b) is defined as the union of all types Φϕ (x, (bj : j ∈ J )) for all ϕ(x, y) ∈ L for which there are (ai0 : i ∈ I ) and (bj0 : j ∈ J ) such that |= ϕ((ai0 : i ∈ I ), (bj0 : j ∈ J )), Ei (ai , ai0 ) for all i ∈ I and Fj (bj , bj0 ) for all j ∈ J . Notice that this is the same thing as tp(a ∗ /b ∗ ) where a ∗ and b ∗ are the hyperimaginaries respectively equivalent to a and b constructed as in the proof of Lemma 15.3. Notice also that a consequence of this definition is that for any sequences of hyperimaginaries c = (ci : i ∈ I ), and a = (ai : i ∈ I ), a ≡b c if and only if a  I0 ≡bJ0 c  I0 for all finite I0 ⊆ I, J0 ⊆ J. Definition 15.13. If B is a set of hyperimaginaries, then tp(a/B) can be defined as tp(a/b) where b is a sequence enumerating B, and a ≡B a 0 can be defined as a ≡b a 0 . If A is a class of hyperimaginaries, a ≡A a 0 means a ≡B a 0 for every subset B ⊆ A. Definition 15.14. Let E be a 0-type-definable equivalence relation. A complete type over a hyperimaginary e (in the variable x) of sort E is a type of the form p(x) = tp(aE /e) where a is a real tuple of the length of x. We say that aE is a realization of p. Of course, p(x) is a partial type over a representative of e but it is complete in the sense that for any aE , bE |= p(x) there is some f ∈ Aut(C/e) such that f(aE ) = bE . After choosing a representative e 0 for e, p(x) can be written as a partial type ð(x, e 0 ) over e 0 . The choice of a different representative e 00 gives rise to an equivalent partial type ð(x, e 00 ) ≡ ð(x, e 0 ) now over e 00 . We denote by SE (e) the set of all complete types over e in the sort E. This notation and terminology can also be used for types over sets of hyperimaginaries. It also makes sense to talk of global types p(x) ∈ SE (C) of hyperimaginary sort E. They can be seen as realized types tp(a/C) in an elementary extension of the monster model C or just as unions ofSa chains of types pi (x) ∈ SE (Ai ) for a family of sets (Ai : i ∈ On) with C = i∈On Ai . Sometimes we say that a type p(x) ∈ SE (e) is of hyperimaginary sort to distinguish it from the case of a type p(x) ∈ S(e) of real sort. Proposition 15.15. For any hyperimaginary e, the equivalence relation x ≡e y (on real tuples of a given length) is type-definable over any representative of e. Proof. If e = aE , then x ≡e y ⇔ ∃u(E(a, u) ∧ xa ≡ yu). a Remark 15.16. Note that the proof of Proposition 15.15 gives an easy proof of the finitary character of ≡e for any hyperimaginary e since for all real tuples a, b: a ≡e b if and only if a 0 ≡e b 0 for all corresponding finite subtuples a 0 of a and b 0 of b.

15. Hyperimaginaries

105

Remark 15.17. Let a be a real tuple, let e be a hyperimaginary, and let p(x) = tp(a/e). For any 0-type-definable equivalence relation E, tp(a/aE e) ≡ p(x) ∧ E(x, a). a

Proof. Easy to check.

Proposition 15.18. For any set of hyperimaginaries A there are hyperimaginaries a, b such that bdd(A) = dclheq (a) and aclheq (A) = dclheq (b). Proof. By Lemma 15.4 bdd(A) = dclheq (B) if B is the class of all hyperimaginaries in bdd(A) of length ≤ ù. For each α ≤ ù there are at most 2|T | many 0-type-definable equivalence relations on α-sequences. For each such equivalence relation E there is an upper bound κE for the number of hyperimaginaries eE in B: there are at most 2|T |+|A| possibilities for p(x) = tp(e/A) and for each such p(x) there are boundedly many d |= p with dE ∈ B. If κ is the supremum of all these κE , it follows that |B| ≤ κ + 2|T |+|A| and we can choose a sequence c enumerating B. By Lemma 15.3, c ∼ b for some hyperimaginary b. Clearly dclheq (b) = dclheq (B) = bdd(A). The case aclheq (A) is similar. a Remark 15.19. bdd is a closure operator on subclasses of C, that is, for all subclasses A, B of C: 1. A ⊆ bdd(A). 2. If A ⊆ B, then bdd(A) ⊆ bdd(B). 3. bdd(bdd(A)) ⊆ bdd(A). Proof. Only 3 needs some checking. We may assume A is a set of hyperimaginaries. By Proposition 15.18 there is some hyperimaginary e ∼ bdd(A), and it suffices to prove that bdd(e) ⊆ bdd(A). Note that e ∈ bdd(A) and let κ be an upper bound for the number of A-conjugates of e. Assume aE ∈ bdd(e) and let ë be an upper bound for the number of e-conjugates of aE . It follows that aE has at most ë · κ conjugates over A and therefore aE ∈ bdd(A). a Lemma 15.20. For any A-hyperimaginary e, there is some hyperimaginary e 0 such that Aut(C/e 0 ) = {f ∈ Aut(C/A) : f(e) = e}, that is, e 0 ∼ Ae. If e is A-bounded, e 0 is A-bounded too. Proof. Let e = bE where E is a type-definable over A equivalence relation. Let a enumerate A, let p(u) = tp(a), and let E = E(x, y; a). We define F (xz, yu) ⇔ (z = u ∧ p(u) ∧ E(x, y; z)) ∨ xz = yu. It is a 0-type-definable equivalence relation. It is easy to see that e 0 = baF is as required. a KP

Proposition 15.21. a ≡A b if and only if a ≡bdd(A) b. Proof. Consider first the case A = ∅. By Proposition 15.18 we can assume bdd(∅) is a single hyperimaginary. The equivalence relation E(a, b) ⇔

106

15. Hyperimaginaries

tp(a/bdd(∅)) = tp(b/bdd(∅)) is bounded and by Proposition 15.15 it is typedefinable over any representative. Since it is invariant, it is also type-definable KP over ∅ and hence ≡⊆ E. For the other direction, assume E(a, b). Note that e = [a]KP is a bounded hyperimaginary and thus e ∈ bdd(∅). Hence there is ≡

KP

some f ∈ Aut(C/e) such that f(a) = b, which implies a ≡ b. The general case cannot be obtained by simply applying the case just proven to T (A) since bdd(A) is the class of all A-bounded hyperimaginaries while bdd(∅) computed in T (A) is the class of all A-bounded A-hyperimaginaries. But Lemma 15.20 helps to solve this difficulty. a KP

KP

KP

Remark 15.22. ≡A is a finitary relation: a ≡A b if and only if a 0 ≡A b 0 for all corresponding finite subtuples a 0 of a and b 0 of b. Proof. By Proposition 15.21, Proposition 15.18, and Remark 15.16.

a

Lemma 15.23. For any 0-type-definable equivalence relation E, the following are equivalent: 1. aE ∈ dclheq (M ). 2. aE ∈ bdd(M ). 3. E(x, a) is finitely satisfiable in M . Proof. Clearly 1 implies 2. We will show 2 ⇒ 3 and 3 ⇒ 1. Assume first that some formula ϕ(x, a) ∈ E(x, a) is not satisfiable in M . For every cardinal κ we can build a coheir sequence over M , (ai : i < κ), starting with a0 = a. If i < j < κ, then |= ¬ϕ(ai , aj ) since otherwise, by indiscernibility, |= ϕ(a, aj ) and hence ϕ(x, a) would be satisfiable in M . Since κ can be arbitrarily large and the elements of the coheir sequence have the same type over M and have different E-classes, aE 6∈ bdd(M ). For 3 ⇒ 1, assume E(x, a) is finitely satisfiable in M and let f ∈ Aut(C/M ) and ϕ(x, y) ∈ E(x, y). We will show that |= ϕ(a, f(a)). This will imply E(a, f(a)) and therefore f(aE ) = aE . We may assume that E(x, y) is closed under conjunction and hence ` ø(x, z) ∧ ø(z, y) → ϕ(x, y) for some symmetric ø(x, y) ∈ E(x, y). Since ø(x, a) is satisfiable in M , there is some c ∈ M such that |= ø(c, a). Since a ≡M f(a), we also have |= ø(c, f(a)). From this it follows that |= ϕ(a, f(a)). a T heq Proposition 15.24. bdd(b) = b∈dclheq (M ) dcl (M ). Proof. If a ∈ bdd(b) and b ∈ dclheq (M ), then clearly a ∈ bdd(M ) and, by Lemma 15.23, we conclude a ∈ dclheq (M ). For the other direction, let us choose a model M such that b ∈ dclheq (M ) (for instance, a model containing a representative of b) and let us choose a cardinal κ > 2|T |+|M | . If a 6∈ bdd(b), there is a family (ai : i < κ) of different b-conjugates of a starting with a0 = a. By choice of κ there are i < j < κ such that ai ≡M aj . Hence for some a 0 6= a we have a ≡M a 0 , which implies a 6∈ dclheq (M ). a

107

15. Hyperimaginaries Proposition 15.25. Let p(x) ∈ S(A). Ls

Ls

1. The restriction ≡A  p of ≡A to p(C) is the finest bounded A-invariant equivalence relation on realizations of p. KP KP 2. The restriction ≡A  p of ≡A to p(C) is the finest bounded type-definable over A equivalence relation on realizations of p. Ls

Proof. Since ≡A  p is bounded and A-invariant, it contains the finest KP bounded A-invariant equivalence relation E on p(C). Similarly, ≡A  p contains the finest bounded type-definable over A equivalence relation F on p(C). On the other hand, E(x, y) ∨ (¬p(x) ∨ ¬p(y)) Ls

is bounded and A-invariant, and therefore it contains ≡A . The corresponding KP result with respect to ≡A and F is more involved. Assume a, b realize p(x) and KP a ≡A b. We will show that F (a, b). By Proposition 15.21 a ≡bdd(A) b. Choose with Lemma 15.5 an extension F 0 of F which is an equivalence relation on all sequences of the length of a, is type-definable over A, and agrees with F on p. Then e = [a]F = [a]F 0 is an A-bounded A-hyperimaginary. By Lemma 15.20 there is an A-bounded hyperimaginary e 0 such that Aut(C/e 0 ) = {f ∈ Aut(C/A) : f(e) = e}. Since e 0 ∈ bdd(A), a ≡e 0 b, which implies that f(a) = b for some f ∈ Aut(C/A) such that f(e) = e and therefore implies that F (a, b). a Proposition 15.26. If E is a bounded equivalence relation on realizations of p(x) ∈ S(A) and it is type-definable over A, then there exists a bounded equivalence relation F defined for all tuples of the length of x which is typedefinable over A and agrees with E in p(C). KP

Proof. Since ≡A  p ⊆ E, it suffices to set F (x, y) ⇔ (p(x) ∧ p(y) ∧ KP E(x, y)) ∨ x ≡A y. a Proposition 15.27. Any A-bounded A-hyperimaginary is an equivalence class of a bounded type-definable over A equivalence relation. Proof. Let E be a type-definable over A equivalence relation and let aE be an A-bounded A-hyperimaginary. Let p(x) = tp(a/A) and note that each KP A-conjugate of aE is an E-class which is also a union of ≡A -classes. Hence if F is defined by KP

F (x, y) ⇔ ∃z(p(z) ∧ E(x, z) ∧ E(y, z)) ∨ x ≡A y, then F is a bounded equivalence relation which is type-definable over A and aE = aF . a Proposition 15.28. Let e, d be hyperimaginaries such that e ∈ bdd(d ) and let A be the set of all d -conjugates of e. There is some hyperimaginary c such that Aut(C/c) = {f ∈ Aut(C) : f(A) = A}.

108

15. Hyperimaginaries

Proof. Let e = aE and d = bG . We may assume every ϕ(x, y) ∈ E(x, y) is symmetric. Since e ∈ bdd(d ), for any such ϕ(x, y) there is a maximal nϕ < ù for which there is a sequence (ai : i < nϕ ) such that de ≡ d [ai ]E for each i < nϕ and |= ¬ϕ(ai , aj ) for all i < j < nϕ . Fix a witnessing sequence (aiϕ : i < nϕ ). Let p(z, x) = tp(d, e), let rϕ (z, xi )i
and let us define F (z1 , z2 ) by ^ ∃(xi : i < nϕ )(rϕ (z1 , xi )i
Note that F is independent of the choice of representatives z1 , z2 in Gclasses. Claim 1. For any f ∈ Aut(C), |= F (b, f(b)) if and only if f(A) = A. Proof of Claim 1. Assume first f(A) = A. Therefore f −1 (A) = A. Clearly, |= rϕ (b, aiϕ )i
Chapter 16

HYPERIMAGINARY FORKING

Definition 16.1. Let A be a class of hyperimaginaries and let I be a set linearly ordered by <. The sequence of hyperimaginaries (ei : i ∈ I ) is indiscernible over A or it is A-indiscernible if for every n < ù, for every two increasing sequences of indices i0 < · · · < in and j0 < · · · < jn , ei0 , . . . , ein ≡A ej0 , . . . , ejn . If A is a set, in practice we may always assume that A is a single hyperimaginary. Note that all the hyperimaginaries ei are in fact of the same sort and hence we can write ei = [ai ]E for a single E. Lemma 16.2. Let d be a hyperimaginary. 1. Let I, J be linearly ordered infinite sets. If (ei : i ∈ I ) is a d -indiscernible sequence of hyperimaginaries, then there is a d -indiscernible sequence (ci : j ∈ J ) such that for every n < ù, for every two increasing sequences of indices i0 < · · · < in ∈ I and j0 < · · · < jn ∈ J , ei0 , . . . , ein ≡d cj0 , . . . , cjn . 2. If (ei : i ∈ I ) and (di : i ∈ I ) are d -indiscernible sequences of hyperimaginaries and (ei : i ∈ I0 ) ≡d (di : i ∈ I0 ) for each finite subset I0 ⊆ I , then f((ei : i ∈ I )) = (di : i ∈ I ) for some f ∈ Aut(C/d ). Proof. 1. By compactness. For 2 note that it follows that (ei : i ∈ I ) ≡d (di : i ∈ I ) (see Remark 15.12). a Proposition 16.3. If (ei : i ∈ I ) is a sequence of hyperimaginaries indiscernible over the hyperimaginary d , then for some representative dˆ of d , some sequence (eˆi : i ∈ I ) of corresponding representatives of (ei : i ∈ I ) is dˆindiscernible. Proof. Fix d 0 , a representative of d . Since the sequence we seek is just a realization of some partial type over d 0 and representatives of the hyperimaginaries ei , we may assume (I, <) = (ù, <). Let κ be an infinite cardinal number larger than the length of d 0 , and larger than the length of every representative of ei , and let ë = i(2κ )+ . By Lemma 16.2 we can extend (ei : i < ù) to a d -indiscernible sequence (ei : i < ë). Choose corresponding representatives [ai ]E = ei . By Proposition 1.6 there is a d 0 -indiscernible sequence (ci : i < ù) such that for all n < ù there are some i0 < · · · < in < ë such 109

110

16. Hyperimaginary forking

that c0 . . . cn ≡d 0 ai0 . . . ain . Since ([ci ]E : i < ù) ≡d (ei : i < ù), for some representative dˆ of d there exists a dˆ-indiscernible sequence (eˆi : i < ù) such that [eˆi ]E = ei . a Proposition 16.4. Let d be a hyperimaginary. 1. For every hyperimaginary e 6∈ bdd(d ), there is a d -indiscernible sequence (ei : i < ù) of distinct hyperimaginaries starting with e0 = e. 2. If the sequence of hyperimaginaries (ei : i ∈ I ) is d -indiscernible, then it is also indiscernible over bdd(d ). ˝ Proof. 1. Let e = aE . Since e 6∈ bdd(d ), by Erdos–Rado (see the proof of Remark 9.2), for some ϕ(x, y) ∈ E(x, y) there are (ai : i < ù) such that a = a0 ≡d ai and |= ¬ϕ(ai , aj ) for all i < j < ù. By ordinary methods, like Ramsey’s Theorem and compactness, we find a d -indiscernible sequence (bi : i < ù) such that |= ¬ϕ(bi , bj ) for all i < j < ù and b0 ≡d a. Then a d -conjugate of ([bi ]E : i < ù) satisfies the requirements. 2. By Proposition 16.3 some sequence (eˆi : i ∈ I ) of representatives is indiscernible over some representative dˆ of d . By Corollary 1.7 (eˆi : i ∈ I ) is indiscernible over some model M containing dˆ. By Proposition 15.24, bdd(d ) ⊆ dclheq (M ) and hence (eˆi : i ∈ I ) and (ei : i ∈ I ) are indiscernible over bdd(d ). a Definition 16.5. The formula ϕ(x, a) divides over the hyperimaginary e (with respect to k) if there is some e-indiscernible sequence (ai : i < ù) with a0 = a for which {ϕ(x, ai ) : i < ù} is inconsistent (k-inconsistent). The formula ϕ(x, a) forks over e if there are ø1 (x, b1 ), . . . , øn (x, bn ) such that ϕ(x, a) ` ø1 (x, b1 ) ∨ · · · ∨ øn (x, bn ) and each øi (x, bi ) divides over e. The set of formulas ð(x) divides ( forks) over e if ð(x) implies some formula which divides (forks) over e. The hyperimaginary a is independent of the hyperimaginary b over the hyperimaginary e (written a ^ | e b) if tp(a/be) (as a ˆ partial type over representatives b, eˆ of b, e) does not fork over e. Note that this ˆ eˆ because equivalent partial is independent of the choice of representatives b, types can be interchanged for dividing and forking purposes. Other notions can be defined in a similar way and we will make use of them when necessary. For instance, a sequence of hyperimaginaries (ei : i ∈ I ) is a Morley sequence over the hyperimaginary e if it is e-indiscernible and ei ^ | e e
16. Hyperimaginary forking

111

Proposition 16.7. A partial type ð(x) divides over the hyperimaginary e with respect to k if and only if it divides over some representative of e with respect to k. Proof. By Proposition 16.3. a Remark 16.8. Let e ∼ e 0 be equivalent hyperimaginaries. A partial type divides over e if and only if it divides over e 0 . The same is true for forking. Proof. Obviously, it is enough to check it for dividing; this is clear, since a sequence is e-indiscernible if and only if it is e 0 -indiscernible. a Remark 16.9. Let a, b, c be hyperimaginaries and assume b ∼ b 0 and c ∼ c 0 . If a ^ | b c, then a ^ | b0 c 0 . Proof. By Remark 16.8, since tp(a/bc) and tp(a/b 0 c 0 ) are equivalent types. a Remark 16.10. Let ð(x, y) be a partial type over ∅. Then ð(x, b) divides over the hyperimaginary e if and only if for some e-indiscernible sequence (bi : i < ù) S with b = b0 , i<ù ð(x, bi ) is inconsistent. Proof. By Proposition 16.7, Proposition 16.3, and Remark 4.2. a Lemma 16.11. For all hyperimaginaries a, b, e: tp(a/be) does not divide over e if and only if for every e-indiscernible sequence I 3 b there is some a 0 ≡eb a such that I is ea 0 -indiscernible. Proof. We adapt the proof of Lemma 4.7. From right to left it is the same as the proof of Lemma 4.7. For the other direction, assume tp(a/be) does not divide over e and, to simplify notation, let I = ([bi ]E : i < ù) be eindiscernible with b = [b0 ]E . By Proposition 16.3 we may assume that (bi : i < ˆ b0 ) = tp(a/eb) ù) is indiscernible over some representative eˆ of e. Let ð(x, e, ˆ bi )i<ù be the set of formulas expressing that (bi : i < ù) is and let Γ(x, e, ˆ ˆ b0 ) ∪ Γ(x, e, ˆ bi )i<ù indiscernible over ex. It is enough S to show that ð(x, e, ˆ bi ) is consistent and can be is consistent. By the above remark, i<ù ð(x, e, ˆ bi )i<ù be a finite subset of Γ(x, e, ˆ bi )i<ù . realized by some c. Let Γ0 (x, e, By Ramsey’s Theorem, there is a one-to-one mapping f : ù → ù such that ˆ f(bi ))i<ù . Now take some c 0 such that c 0 (bi : i < ù) ≡eˆ c(f(bi ) : |= Γ0 (c, e, ˆ bi )i<ù and ð(x, e, ˆ b0 ). i < ù) and note that c 0 realizes Γ0 (x, e, a Proposition 16.12. For all hyperimaginaries a, b, c, d : if tp(b/cd ) does not divide over d and tp(a/cbd ) does not divide over bd , then tp(ab/cd ) does not divide over d . Proof. By Lemma 16.11. a Proposition 16.13. 1. Let ð(x) be a partial type over A. If ð does not fork over the hyperimaginary e, then some completion p(x) ∈ S(A) of ð does not fork over e. 2. Let a, b, c be hyperimaginaries such that a ^ | b c. Then for every hyperimaginary d , there is some a 0 ≡bc a such that a 0 ^ | b cd .

112

16. Hyperimaginary forking

Proof. 1. As in Remark 4.4, the reason is that ð(x) ∪ {¬ϕ(x) : ϕ(x) ∈ L(A) forks over e} is consistent. ˆ c, ˆ c). ˆ dˆ of b, c, d and put tp(a/bc) = ð(x, b, ˆ By 2. Fix representatives b, ˆ c) ˆ which does not fork 1 there exists a completion p(x) ∈ S(bˆ cˆ dˆ) of ð(x, b, over b. Let a be of sort E, let aˆ |= p and let a 0 = aˆE . Then a 0 ≡bc a. Since tp(a 0 /bcd ), as a partial type over bˆ cˆ dˆ, is contained in p, a 0 ^ | b cd . a Proposition 16.14. Let e be a hyperimaginary. 1. If ð(x) divides over e and ð(x) is a partial type over A, then ð(x) divides over eˆ for any representative eˆ of e such that eˆ ^ | e A. 2. If T is simple, then a ^ | e e for any hyperimaginary a. 3. If T is simple, then a partial type ð(x) forks over e if and only if ð(x) forks over some representative of e. Proof. 1. Fix ϕ(x, y) ∈ L and b ∈ A such that ð(x) ` ϕ(x, b) and ϕ(x, b) divides over e. Then for some e-indiscernible sequence (bi : i < ù) with b = b0 , {ϕ(x, bi ) : i < ù} is inconsistent. Since eˆ ^ | e b, by Lemma 16.11 ˆ there is another e-indiscernible sequence (bi0 : i < ù) with b = b00 and such ˆ that {ϕ(x, bi0 ) : i < ù} is inconsistent. Then ϕ(x, b) divides over e. 2. Choose a representative eˆ of e. We will check that the partial type ˆ = tp(a/e) does not fork over e. Assume ð(x, e) ˆ ` ϕ1 (x, a1 ) ∨ · · · ∨ ð(x, e) ϕn (x, an ) where every ϕi (x, ai ) divides over e with respect to ki . Let k = ˆ ∆, max{k1 , . . . , kn }, let ∆ = {ϕ1 (x, y1 ), . . . , ϕn (x, yn )}, and let m = D(ð(x, e), ˆ 1 , . . . , an ) of ð(x, e) ˆ k). By Proposition 3.12 there is a completion p(x) ∈ S(ea with D(p(x), ∆, k) = m. For some i, ϕi (x, ai ) ∈ p. Now, ϕi (x, ai ) divides over e with respect to k and by Proposition 16.7 it divides over some represenˆ ≡ ð(x, e). ˜ Then D(ð(x, e) ˜ ∪ tative e˜ of e with respect to k. Notice that ð(x, e) ˆ ∆, k) = D(ð(x, {ϕi (x, ai )}, ∆, k) ≥ D(p(x), ∆, k) = m and hence D(ð(x, e), ˜ ∆, k) ≥ m + 1, a contradiction. e), 3. Assume ð(x) ` ϕ1 (x, a1 ) ∨ · · · ∨ ϕn (x, an ) where every ϕi (x, ai ) divides over e. By 2 and Proposition 16.13 we can choose a representative eˆ of e such ˆ Hence ð(x) forks that eˆ ^ | e a1 , . . . , an . By 1 every ϕi (x, ai ) divides over e. ˆ over e. a Corollary 16.15. If T is simple, a partial type forks over a hyperimaginary e if and only if it divides over e. Proof. By Proposition 16.14 and Proposition 5.17. a Proposition 16.16. Let T be simple. For all hyperimaginaries a, b, c, d : 1. If ab ^ | c d , then a ^ | c d and a ^ | bc d . 2. If a ^ | b cd , then a ^ | b d and a ^ | bc d . 3. There is some a 0 ≡b a such that a 0 ^ | b c. 4. If a ^ | c d and b ^ | ac d , then ab ^ | c d. 5. a ^ | b bdd(b).

16. Hyperimaginary forking

113

6. If d ^ | c ab, then a ^ | c b if and only if a ^ | cd b. Proof. 1 and 2 follow straightforwardly from Corollary 16.15 and the definition of dividing. To check that a ^ | bc d in 1 and 2 it may be convenient to use Remark 16.10 applied to tp(a/bcd ). 3 follows from point 2 in Proposition 16.14 and Proposition 16.13, while 4 follows from Proposition 16.12 and Corollary 16.15. 5. By 3 there is some a 0 ≡b a such that a 0 ^ | b bdd(b). There is some 0 f ∈ Aut(C/b) such that f(a ) = a. Since f fixes setwise bdd(b), if we apply f we obtain a ^ | b bdd(b). 6. Assume d ^ | c ab and a ^ | c b. By 2 d ^ | ac b and then by 4 da ^ | c b. By 1 a^ | cd b. Assume now d ^ | c ab and a ^ | cd b. By 2 d ^ | c b and by 4, ad ^ | c b. Then by 1 a ^ | c b. a Lemma 16.17. Let T be simple. For any hyperimaginaries a, b, c the following are equivalent: 1. a ^ | b c. ˆ cˆ of a, c such that aˆ ^ ˆ 2. There are representatives a, | b c. ˆ ˆ 3. a ^ | bˆ c for every representative b of b such that b ^ | b ac. ˆ ˆ 4. a ^ | bˆ c for some representative b of b such that b ^ | b ac. Proof. 1 ⇒ 2. We use the different points of Proposition 16.16 several times. By point 3 of 16.16 there is some representative aˆ of a such that ˆ cˆ be representatives of aˆ ^ | a bc. By points 4 and 1 of 16.16, aˆ ^ | b c. Let b, ˆ c). ˆ ˆ This partial type does not fork over b and b, c and write tp(a/bc) = ð(x, b, ˆ by Proposition 16.13 it can be extended to some complete type p(x) ∈ S(bˆ c) ˆ Since p ` tp(a1 /bˆ cb), ˆ which does not fork over b. Let a1 |= p. Then a1 ≡bc a. ˆ Let c1 be such that a1 cˆ ≡bc ac ˆ 1 . Then a1 ^ | b bˆ cˆ and, in particular, a1 ^ | b c. aˆ ^ | b c1 and c1 is also a representative of c. ˆ c) ˆ (as a partial type over the real tuple cˆ and a 2 ⇒ 1. Clear since tp(a/b ˆ ˆ b). representative bˆ of b) includes tp(a/bc) (as a partial type over c, 1 ⇒ 3. Assume a ^ | b c and bˆ ^ | b ac. By point 6 of Proposition 16.16, ˆ ˆ a^ | b bˆ c. Since b b ∼ b, by Remark 16.9, a ^ | bˆ c. ˆ 3 ⇒ 4. Clear, since there are such b. 4 ⇒ 1. Assume a ^ | bˆ c and bˆ ^ | b ac. By Remark 16.9 a ^ | b bˆ c and by point 6 of 16.16, a ^ | b c. a Proposition 16.18 (Symmetry and transitivity). If T is simple, then independence is symmetric and transitive for hyperimaginaries, that is, for any hyperimaginaries a, b, c, d : 1. a ^ | b c if and only if c ^ | b a. 2. If a ^ | b c and a ^ | bc d , then a ^ | b cd .

114

16. Hyperimaginary forking

Proof. 2 is a consequence of 1 and of point 4 of Proposition 16.16. For 1, use Lemma 16.17 and symmetry of independence for real tuples. a Proposition 16.19. Let T be simple. For all hyperimaginaries a, b, c: 1. a ^ | b c if and only if a ^ | b bdd(c) if and only if a ^ | bdd(b) c if and only if bdd(a) ^ | b c. 2. a ^ | b a if and only if a ∈ bdd(b). Proof. 1 follows easily from point 5 of Proposition 16.16 and the basic properties of independence. Similarly for the direction from right to left of 2. For the other direction, assume a 6∈ bdd(b). By Proposition 16.4 there is a bindiscernible sequence (ai : i < ù) with a0 = a and ai 6= aj for all i < j < ù. Let a be of sort E and choose with Proposition 16.3 a b-indiscernible sequence S (aˆi : i < ù) of representatives. Since i<ù E(x, aˆi ) is inconsistent, the partial type E(x, aˆ0 ) divides and forks over b. Hence a ^ 6 | b a. a Proposition 16.20 (Finite character). Let T be simple. Let a, b hyperimaginaries and let c = (ci : i ∈ I ) a sequence of hyperimaginaries. If for each finite I0 ⊆ I , a ^ | b (ci : i ∈ I0 ), then a ^ | b c. Proof. Let bˆ be a representative of b and let (cˆi : i ∈ I ) be a corresponding sequence of representatives for (ci : i ∈ I ). Write tp(a/b(ci : i ∈ I )) as a ˆ (cˆi : i ∈ I )). For each finite subset Σ of this partial type partial type ð(x, b, we can find some finite I0 ⊆ I such that tp(a/b(ci : i ∈ I0 )) can be written ˆ cˆi : i ∈ I0 ) containing Σ. Therefore Σ does not fork as a partial type over b( ˆ (cˆi : i ∈ I )) does not fork over b and a | c. over b. Then ð(x, b, a ^b Proposition 16.21 (Local character). Let T be simple, let a be a hyperimaginary of length ë and let b = (bi : i ∈ I ) be a sequence of hyperimaginaries. Then a ^ | (b :i∈J ) b for some J ⊆ I such that |J | ≤ |T | + ë. i

Proof. We may assume I = κ for some cardinal κ. Choose inductively representatives bˆ i of bi such that bˆ i ^ | b b bˆ
0

i

0

Corollary 16.22. Let T be simple. For any hyperimaginaries a, b, if a has length ≤ ë, there is some hyperimaginary e of length ≤ |T | + ë such that e ∈ dclheq (b) and a ^ | e b. Proof. By Lemma 15.4 there is a sequence (bi : i ∈ I ) of countable hyperimaginaries bi such that b ∼ (bi : i ∈ I ). By Proposition 16.21 there is some J ⊆ I such that |J | ≤ |T | + ë and a ^ | (b :i∈J ) (bi : i ∈ I ). Then i e = (bi : i ∈ J ) satisfies the requirements. a

16. Hyperimaginary forking

115

Proposition 16.23. Let T be simple, let e be a hyperimaginary, and let ð(x, y) be a set of formulas over ∅. Then ð(x, S a) divides over e if and only if for any Morley sequence (ai : i < ù) in tp(a/e), i<ù ð(x, ai ) is inconsistent. Proof. It is an adaptation of the proof of Proposition 5.15. We may assume ð(x, y) = {ϕ(x, y)}. Let (ai : i < ù) be a Morley sequence in tp(a/e) and assume ϕ(x, a) divides over e and {ϕ(x, ai ) : i ∈ I } is consistent. Let κ ≥ |T | be larger than the length of e and let (I, <) be a linearly ordered set isomorphic to the reverse order of κ + . There is a Morley sequence aI = (ai : i < κ) in tp(a/e) such that {ϕ(x, ai ) : i ∈ I } is consistent. Let c realize this set of formulas. By Proposition 16.21 there is some J ⊆ I of cardinality ≤ κ such that c ^ | ea aI . We can find i ∈ I such that i < J . Note that Lemma 5.14 is J valid over a hyperimaginary and therefore aJ ^ | e ai . Proposition 4.9 is also valid over a hyperimaginary and hence ϕ(x, ai ) divides over eaJ . But then c^ 6 | ea aI , a contradiction. a J

Proposition 16.24 (Independence Theorem). Let T be simple, and let a, b, c, d be hyperimaginaries such that a ^ | M b, c ^ | M a, d ^ | M b, and c ≡M d . Then there is some hyperimaginary e ^ | M ab such that e ≡Ma c and e ≡Mb d . Proof. We may assume that c and d are real tuples (replace c by a representative cˆ such that cˆ ^ | c Ma and then replace d by some representative dˆ ˆ bˆ of a and b such such that cˆ ≡M dˆ and dˆ ^ | Md b). Choose representatives a, ˆ Consider tp(c/aM ) and tp(d/bM ) as partial types over M aˆ that aˆ ^ | M b. ˆ ˆ and M b respectively. They can be extended to complete types p(x) ∈ S(M a) ˆ which do not fork over M . Note that p  M = q  M . and q(x) ∈ S(M b) By the Independence Theorem for ordinary types there is some eˆ |= p ∪ q ˆ Then eˆ is a representative of the hyperimaginary e we are such that eˆ ^ | M aˆ b. seeking. a Definition 16.25. SU-rank can be extended to hyperimaginaries in a natural way by the following conditions: 1. SU(a/e) ≥ 0. 2. SU(a/e) ≥ α + 1 if and only if SU(a/eb) ≥ α for some hyperimaginary b such that a ^ 6 | e b. 3. SU(a/e) ≥ α if and only if SU(a/e) ≥ â for all â < α, if α is a limit ordinal. One can check that {α : SU(a/e) ≥ α} is an initial segment of the class of all ordinals numbers. We define SU(a/e) = sup{α : SU(a/e) ≥ α}. Hence, SU(a/e) = ∞ if and only if SU(a/e) ≥ α for all ordinals α. Remark 16.26. Let T be simple. For all hyperimaginaries a, b, e: 1. SU(a/e) ≥ SU(a/eb). 2. If a ^ | e b, then SU(a/e) = SU(a/eb).

116

16. Hyperimaginary forking

3. If SU(a/e) = SU(a/eb) < ∞, then a ^ | e b. 4. If SU(ab/e) < ∞, then SU(a/eb) + SU(b/e) ≤ SU(ab/e) ≤ SU(a/eb) ⊕ SU(b/e). 5. If b ∈ bdd(ae), then SU(ab/e) = SU(a/e). Proof. It is just a straightforward adaptation of the proofs of propositions 13.11, 13.16, and 13.18. a Proposition 16.27. Let T be simple. For all hyperimaginaries a, e: 1. If T is supersimple and a is finitary, then SU(a/e) < ∞. 2. SU(a/e) = 0 if and only if a ∈ bdd(e). 3. SU(a/e) = ∞ if and only if there is a sequence (ai : i < ù) of hyperimaginaries ai such that a ^ 6 | ea ai for all i < ù.
a ≡e b if and only if Lstp(a/e) = Lstp(b/e). Remark 16.29. For any hyperimaginaries a, b, e: Ls Ls ˆ bˆ of a, b. 1. a ≡e b if and only if aˆ ≡e bˆ for some representatives a, heq 2. If a ≡M b and e ∈ dcl (M ) then for some hyperimaginary c, there are infinite e-indiscernible sequences I, J , such that a, c ∈ I and c, b ∈ J .

16. Hyperimaginary forking

117

3. If a, b ∈ I for some infinite e-indiscernible sequence I , then a ≡M b for some model M such that e ∈ dclheq (M ). Ls 4. a ≡e b if and only if for some n < ù there are infinite e-indiscernible sequences Ii and hyperimaginaries ai such that a = a0 , ai , ai+1 ∈ Ii and an+1 = b. ˆ bˆ of a, b such that aˆ ≡M Proof. 1 is clear. For 2, take representatives a,

ˆ By point 2 of Lemma 9.12, there is some cˆ and some M -indiscernible b. ˆ cˆ ∈ I and c, ˆ bˆ ∈ J . The (hence e-indiscernible) sequences I, J such that a, corresponding hyperimaginaries and sequences of hyperimaginaries are as required. 3. Let I be an e-indiscernible sequence such that a, b ∈ I . By Propoˆ eˆ of a, b, e such that for some eˆ b, ˆ sition 16.3, there are representatives a, ˆ ˆ ˆ b ∈ J . By point 1 of Lemma 9.12, aˆ ≡M b for indiscernible sequence J , a, ˆ Then e ∈ dclheq (M ) and a ≡M b. some model M 3 e. 4 is a consequence of points 2 and 3. a Definition 16.30. Le e be a hyperimaginary. A relation E is type-definable over e if it is type-definable and e-invariant. Note that this is equivalent to saying that E is type-definable over any representative of e. An equivalence class of a tuple in a type-definable over e equivalence relation is an e-hyperimaginary. Lemma 16.31. Let e be a hyperimaginary. For any e-hyperimaginary h there is a hyperimaginary h 0 such that h 0 ∼ he, that is, Aut(C/h 0 ) = Aut(C/he). Moreover h 0 is e-bounded if h is e-bounded. Proof. It is a generalization of Lemma 15.20, with a similar proof. Let h = bE where E is a type-definable over e equivalence relation. Let eˆ be a ˆ and let E = E(x, y; e). ˆ We representative of e, say eˆG = e, let p(x) = tp(e), define F (xz, yu) ⇔ (G(z, u) ∧ p(z) ∧ p(u) ∧ E(x, y; z)) ∨ xz = yu. It is a 0-type-definable equivalence relation and it is easy to see that h 0 = b eˆF is as required. a Lemma 16.32. Let T be simple. Let a, b, e be hyperimaginaries such that ˆ bˆ of a, b such that aˆ ^ a^ | e b and a ≡e b. There are representatives a, | e bˆ and ˆ aˆ ≡e b. Proof. Start with a representative aˆ of a such that aˆ ^ | a eb. It follows that ˆ ˆ aˆ ^ | e b. Next take some representative b of b such that bˆ ≡e aˆ and bˆ ^ | eb a. ˆ Then aˆ ^ | e b. a

118

16. Hyperimaginary forking

Lemma 16.33. Let T be simple. Let a, b, e be hyperimaginaries such that a ≡M b for some model M such that e ∈ dclheq (M ). Then a ≡N b for some model N such that e ∈ dclheq (N ) and ab ^ | e N. Proof. We may assume that a, b are real tuples. Choose a model M 0 ≡e M such that M 0 ^ | e M and then choose a model N such that N ≡M M 0 and tp(N/Mab) is a coheir of tp(N/M ). Then N ^ | e M and by Proposition 7.6 N^ | M ab. By transitivity, N ^ | e ab. Since tp(N/Mab) does not split over M and a ≡M b, it follows that a ≡N b. a Proposition 16.34. Let T be simple. For any hyperimaginaries a, b, e such Ls that a ≡e b and a ^ | e b there is some model M such that a ≡M b, ab ^ | e M and e ∈ dclheq (M ). Moreover a, b ∈ I for some infinite M -indiscernible sequence I . Proof. Fix n < ù, fix models Mi for i ≤ n and sequences ai for i ≤ n + 1 such that e ∈ dclheq (Mi ) and a = a0 ≡M0 a1 ≡M1 a2 ≡M3 · · · ≡Mn an+1 = b. By Lemma 16.33 we may assume that ai ai+1 ^ | e Mi . Hence we can also assume that Mi ^ | e a0 , . . . , an+1 (Mj : j < i). It follows that a0 , . . . , an+1 ^ | M0 , . . . , Mn . e

Note that a0 ^ | M ,...,M an+1 . Let b0 = a0 , let bn+1 = an+1 and for 1 ≤ i ≤ n n 0 choose bi ≡M0 ,...,Mn ai and such that bi ^ | M ,...,M bn+1 (bj : j < i) for any n 0 i ≤ n. It follows that (bj : j ≤ n + 1) is independent over M0 , . . . , Mn . Note that a = b0 ≡M0 b1 ≡M1 b2 ≡M3 · · · ≡Mn bn+1 = b. Since bj ^ | e M0 , . . . , Mn , the sequence (bj : j ≤ n + 1) is also independent over e and moreover b0 , . . . , bn+1 ^ | e M0 , . . . , Mn . Hence we can proceed by induction on n. The case n = 0 is clear and using the induction hypothesis it is enough now to consider the case n = 1. We have a = b0 ≡M0 b1 ≡M1 b2 = b. Let f ∈ Aut(C/M0 ) such that f(b1 ) = b0 . Then M1 ^ | M b1 b2 , f(M1 ) ^ | M b0 and b0 ^ | M b1 b2 and by 0 0 0 the Independence Theorem (Proposition 16.24) there is a model N such that N^ | M b0 b1 b2 , N ≡M0 b1 b2 M1 and N ≡M0 b0 f(M1 ). Then b0 N ≡ b0 f(M1 ) ≡ 0 b1 M1 ≡ b2 M1 ≡ b2 N and clearly b0 b2 ^ | e N. The last assertion follows from Proposition 10.11 since a ^ | M b and by Lemma 16.32 we can assume a, b are real tuples. a Proposition 16.35. If T is simple, then for any hyperimaginaries a = aˆF , b = bˆ F , and e, the following are equivalent: Ls

1. a ≡e b.

16. Hyperimaginary forking

119

2. For some hyperimaginary c there are infinite e-indiscernible sequences I, J , such that a, c ∈ I and c, b ∈ J . 3. a ≡bdd(e) b. ˆ for every bounded e-type-definable equivalence relation E ⊇ F . ˆ b) 4. E(a, Proof. By Remark 16.29 it is clear that 2 implies 1 and that 1 implies 3. Ls To prove 1 ⇒ 2 find c such that c ^ | e ab and a ≡e c and then use Proposition 16.34. To find such c one needs to adapt the proof of Lemma 10.2, but this is straightforward. 3 ⇒ 4. aˆE is an e-hyperimaginary and by Lemma 16.31 there is a hyperimaginary c such that c ∼ e, aˆE . Since aˆE is e-bounded, c ∈ bdd(e). By 3, ˆ and ˆ b) there is some f ∈ Aut(C/bdd(e)) such that f(a) = b. Then F (f(a), ˆ ˆ ˆ ˆ b). Since f(c) = c, bE = f(aˆE ) = aˆE and hence E(a, ˆ b). therefore E(f(a), Ls 4 ⇒ 1. Let E(x, y) be defined as xF ≡e yF . The condition in 2 can be ˆ over any representative eˆ of e. Hence expressed by a partial type Φ(x, y, e) ˆ defines E, which is a bounded e-type-definable equivalence relation Φ(x, y, e) Ls ˆ and hence a ≡ ˆ b) E ⊇ F . By 4, E(a, a e b. Corollary 16.36. Let T be simple. Let X be the set of all finite subsets of I and let e = (eJ : J ∈ X ) be a sequence of hyperimaginaries such that Ls eJ ∈ dclheq (eJ 0 ) whenever J ⊆ J 0 . For all hyperimaginaries a, b: a ≡e b if and Ls only if a ≡eJ b for all J ∈ X . Proof. By Proposition 16.35 and compactness.

a

Corollary 16.37 (Independence Theorem for hyperimaginary Lascar strong types). Let T be simple, let a1 , a2 , c1 , c2 , e be hyperimaginaries such that c1 ^ | e c2 , a1 ^ | e c1 , a2 ^ | e c2 and a1 ≡bdd(e) a2 . Then there is some hyperimaginary a ^ | e c1 c2 such that a ≡bdd(e)c1 a1 and a ≡bdd(e)c2 a2 . Ls

Proof. By Proposition 16.35, a1 ≡e a2 and then by Proposition 16.34 a1 ≡M a2 for some model M such that a1 a2 ^ | e M and e ∈ dclheq (M ). We can assume that M ^ | a a e c1 c2 and therefore M ^ | e c1 c2 a1 a2 . It follows 1 2 that c1 ^ | M c 2 , a1 ^ | M c1 and a2 ^ | M c2 . By Proposition 16.24 there is some a^ | M c1 c2 such that a ≡Mc1 a1 and a ≡Mc2 a2 . Clearly a ^ | e c1 c2 , a ≡bdd(e)c1 a1 and a ≡bdd(e)c2 a2 . a Corollary 16.38. Let T be simple, let e be a hyperimaginary, and let ϕ(x, y), ø(x, z) ∈ L. Assume ϕ(x, a) ∧ ø(x, b) does not fork over e. If Ls b ≡e b 0 , a ^ | e b, and a ^ | e b 0 , then ϕ(x, a) ∧ ø(x, b 0 ) does not fork over e. Proof. Choose c ^ | e ab such that c |= ϕ(x, a) ∧ ø(x, b). Then a ^ | e c, Ls b0 ^ | e a, b ^ | e c, and b ≡e b 0 . We can apply the Independence Theorem 16.37 thus obtaining some b 00 ^ | e ac such that b 00 ≡ae b 0 and b 00 ≡ce b. We choose

120

16. Hyperimaginary forking

now c 0 such that b 00 c ≡ae b 0 c 0 . It is easy to see that c 0 ^ | e ab 0 and c 0 |= 0 ϕ(x, a) ∧ ø(x, b ). a Corollary 16.39. Let T be simple, let Σ(x) be a partial type representing a complete type over a hyperimaginary e, and let ð(x) be a partial type. Then Σ(x) ∪ ð(x) does not fork over e if and only if D(Σ(x), ∆, k) = D(Σ(x) ∪ ð(x), ∆, k) for all finite ∆, k. Proof. It is an easy adaptation of the proof of Proposition 5.22 once we know that forking over hyperimaginaries has all the good properties. a Lemma 16.40. Let ϕ(x, y) ∈ L, and n, k < ù. For any hyperimaginary e, for any ordinals α, â, {(a, b) : a, b are tuples of length α, â and D(tp(a/eb), ϕ, k) ≥ n} is type-definable over any representative of e. Proof. Let eˆ be a representative of e, say e = eˆE . As in the proof of Remark 3.14, D(tp(a/eb), ϕ, k) ≥ n if and only there is a tree (as : s ∈ ù ≤n ) such that for each f ∈ ù n , tp(a/eb)(x) ∪ {ϕ(x, afi+1 ) : i < n} is consistent and for each i < n, for each s ∈ ù i , {ϕ(x, as a j ) : j < ù} is k-inconsistent. Now consistency of tp(a/eb)(x)∪{ϕ(x, afi+1 ) : i < n} can be expressed by: ^ ^ ˆ ∧ ˆ ↔ ø(a, b, z)). ∃x( ϕ(x, afi+1 ) ∧ ∃z(E(z, e) (ø(x, b, e) a i
is type-definable over any representative of e. Proof. For each ϕ(x, y) ∈ L, for each k < ù, let nϕk = D(p(x), ϕ, k). By Corollary 16.39, for any a |= p, a ^ | e b if and only if D(tp(a/eb), ϕ, k) ≥ nϕk for all ϕ, k. The result follows then by Lemma 16.40. a Corollary 16.42. Let T be simple, let e be a hyperimaginary, and fix p(x) ∈ S(e). 1. For any n < ù {(a1 , . . . , an ) : (a1 , . . . , an ) is e-independent and ai |= p for i = 1, . . . , n} is type-definable over any representative of e. 2. For any linearly ordered set I , {(ai : i ∈ I ) : (ai : i ∈ I ) is a Morley sequence in p} is type-definable over any representative of e. Proof. Like the proof of Corollary 5.24, using Lemma 16.40 and Corollary 16.39. a

Chapter 17

CANONICAL BASES REVISITED

Definition 17.1. Let p(x) ∈ SE (e) be a complete hyperimaginary type over the hyperimaginary e, that is, p(x) = tp(a/e) where a, e are hyperimaginaries and E is the sort of a. We say that p(x) is an amalgamation base if the Independence Theorem is true for p(x): for any hyperimaginaries c1 , c2 such that c1 ^ | e c2 , for any hyperimaginaries a1 , a2 such that tp(a1 /e) = p(x) = tp(a2 /e), a1 ^ | e c1 , and a2 ^ | e c2 , there is some hyperimaginary a ^ | e c1 c2 such that a ≡ec1 a1 and a ≡ec2 a2 . It is easy to see that (if T is simple) we can always check this condition assuming c1 , c2 are real tuples enumerating models M1 , M2 such that e ∈ dclheq (Mi ) for i = 1, 2. As shown in Proposition 16.24 and in Corollary 16.37, in a simple theory any type over a model and any type over a hyperimaginary of the form bdd(e) is an amalgamation base. Hence Lascar strong types (more precisely, their corresponding sets of formulas) in simple theories are amalgamation bases. Let p(x) ∈ SE (e), and let d ∈ dclheq (e). By p(x)  d we refer to the type tp(b/d ) where b is an arbitrary realization of p. It is well defined. Note that if q(x) is another complete type of sort E and d is also definable over its domain, then the consistency of p(x) ∪ q(x) implies p  d = q  d . In particular, if p ∈ SE (C) is a global type in the hyperimaginary sort E, then p  e is well defined for any hyperimaginary e. The notion of stationary type also makes sense for hyperimaginary types. p(x) ∈ SE (e) is stationary if there is a unique global type p(x) extending p (i.e., such that p  e = p) that does not fork over e. Remark 17.2. Let T be simple. For any hyperimaginaries a, e, tp(a/e) is an amalgamation base if and only if tp(a/e) ` tp(a/bdd(e)). Proof. Assume tp(a/e) is an amalgamation base and a 0 ≡e a. We will show that a 0 ≡bdd(e) a. Notice that a ^ | e bdd(e), a 0 ^ | e bdd(e), and bdd(e) ^ | e bdd(e). We can amalgamate these types and obtain some a 00 such that a 00 ^ | e bdd(e), a 00 ≡bdd(e) a 0 , and a 00 ≡bdd(e) a. It follows that 0 a ≡bdd(e) a . The other direction follows from Corollary 16.37. a Proposition 17.3. 1. In a simple theory, any stationary type is an amalgamation base. 121

122

17. Canonical bases revisited

2. Let T be simple. Assume p(x) ∈ SE (e) is stationary and q(x) ∈ SE (d ) is an amalgamation base. If p and q have a common nonforking extension, then q is stationary. 3. In a stable theory, all amalgamation bases are stationary. Proof. 1. Given a1 , a2 , c1 , c2 , e and a stationary type p(x) over e as in the definition of amalgamation base, any realization a of the unique nonforking extension of p over e, c1 , c2 satisfies the requirements. 2. Assume there is a set A such that d ∈ dclheq (A) and q has two different nonforking extensions q1 (x), q2 (x) over A. Without loss of generality, e ∈ dclheq (A) and p = q1 . Choose A0 ≡e A such that A0 ^ | e A and let p 0 be the corresponding copy of p over A by an automorphism over e. Then p0 is stationary and a nonforking extension of q. Since q is an amalgamation base, q2 and p 0 have a common global nonforking extension over AA0 . But p and p 0 also have a common nonforking extension over AA0 . By stationarity of p0 they coincide, and therefore p = q2 . 3. Let T be stable. By Lemma 16.17 we only need to consider the case of a type p(x) ∈ S(e) of real variables. If e is a sequence of real parameters the result follows easily from Remark 17.2, since in T strong types are stationary. But, in any case, the general case follows from 2 since in T stable all amalgamation bases have nonforking stationary extensions, for instance nonforking extensions over a model. a Remark 17.4. Let T be simple. Assume a, b, e are hyperimaginaries and b ∈ dclheq (ae). If tp(a/e) is an amalgamation base, then tp(b/e) is also an amalgamation base. Proof. By Remark 17.2. a Definition 17.5. Let p(x) ∈ SE (e) for a hyperimaginary e and let A be a class of hyperimaginaries. We say that p is finitely satisfiable in A if for every formula ϕ(x) ∈ p(x) (considering p(x) as a partial type, which we assume is closed under finite conjunctions, over some representative of e) there is some ˆ for every representative bˆ of bE . hyperimaginary bE ∈ A such that |= ϕ(b) This definition is independent of the chosen representative of e. Proposition 17.6. Let T be simple and let a, d, e be hyperimaginaries. 1. If tp(a/e) is finitely satisfiable in dclheq (d ) and d ∈ dclheq (e), then a ^ | d e.

2. If tp(a/e) is finitely satisfiable in dclheq (e), then tp(a/e) is an amalgamation base. ˆ e) ˆ ∈ tp(a/e) then |= ϕ(b, ˆ Proof. 1. If eˆ is a representative of e and ϕ(x, e) heq ˆ ˆ does not for all representatives b of some b ∈ dcl (d ) and hence ϕ(x, e) divide over d . 2. By Remark 17.2 it suffices to show that tp(a/e) ` tp(a/bdd(e)). We use Proposition 16.35. Let a = aF0 , b = bF0 , let E ⊇ F be a type-definable over

17. Canonical bases revisited

123

e bounded equivalence relation, assume a ≡e b, and let us prove E(a 0 , b 0 ). Choose a representative e 0 of e, and choose some f ∈ Aut(C/e) such that f(a) = b. Then e 00 = f(e 0 ) is also a representative of e. Let Σ(x, y, u) be a partial type over ∅ such that Σ(x, y, e 0 ) defines E. Then Σ(x, y, e 00 ) also defines E. Consider some ϕ(x, y, u) ∈ Σ(x, y, u). We will check that |= ϕ(a 0 , b 0 , e 0 ). Choose now by compactness some symmetric (in x, y) formula ø(x, y, u) ∈ Σ(x, y, u) such that ø(x, y1 , e 0 ) ∧ ø(y1 , y2 , e 0 ) ∧ ø(y2 , y3 , e 0 ) ∧ ø(y3 , y4 , e 00 ) ∧ ø(y4 , z, e 0 ) ` ϕ(x, z, e 0 ). Since E is bounded, ø(x, y, e 0 ) is thick and we can choose a maximal sequence c1 , . . . , cn ∈ dclheq (e) such that |= ¬ø(ci0 , cj0 , e 0 ) for all representatives ci0 , cj0 of ci , cj , for all i 6= j. Assume that |= ø(a 00 , ci0 , e 0 ) for some representative a 00 of a, for some representative ci0 of ci , for some i ≤ n. Then f(a 00 ) is a representative of b, f(ci0 ) is a representative of ci , and |= ø(f(a 00 ), f(ci0 ), e 00 ). Since F (x, y) ` ø(x, y, e 0 ), by choice of ø we obtain that |= ϕ(a 0 , b 0 , e 0 ). Assume now that |= ¬ø(a 00 , ci0 , e 0 ) for any representative a 00 of a, for any representative ci0 of ci , for any i ≤ n. Consider tp(a/ec1 , . . . , cn ) as a partial type over e 0 , c10 , . . . , cn0 for some representatives c10 , . . . , cn0 of c1 , . . . , cn . Notice that it is finitely satisfiable in dclheq (e). By compactness there is some symmetric formula ε(x, y) ∈ F (x, y) such that |= ∀x 0 x1 . . . xn (ε(x 0 , a 0 ) ∧

n ^

ε(xi , ci0 ) →

i=1

n ^

¬ø(x 0 , xi , e 0 )).

i=1

Vn 0 If Vnè(y, z, z10, . . . , zn ) is the formula ∀x x1 . . . xn (ε(x , y) ∧ i=1 ε(xi , zi ) → i=1 ¬ø(x , xi , z)), then, by the way tp(a/ec1 , . . . , cn ) has been constructed, ∃y(ε(x, y) ∧ è(y, e 0 , c10 , . . . , cn0 )) ∈ tp(a/ec1 , . . . , cn ) and therefore there is some d ∈ dclheq (e) of sort F such that |= ∃y(ε(d 0 , y) ∧ è(y, e 0 , c10 , . . . , cn0 )) for every representative d 0 of d . This contradicts the maximality of the sequence c1 , . . . , cn . a 0

Corollary 17.7. Let T be simple. If (ai : i ≤ ù) is an indiscernible sequence of hyperimaginaries, then tp(aù /(ai : i < ù)) is an amalgamation base. Proof. It is a consequence of Proposition 17.6 since tp(aù /(ai : i < ù)) is finitely satisfiable in {ai : i < ù}. a Proposition 17.8. Let T be simple, let aE be a hyperimaginary and let D = dclheq (a)∩bdd(aE ). Then tp(a/D) is an amalgamation base. Moreover there is some 0-type-definable equivalence relation E 0 such that aE 0 is equivalent to some enumeration of D and hence aE 0 ∈ bdd(aE ), aE ∈ dclheq (aE 0 ), and tp(a/aE 0 ) is an amalgamation base.

124

17. Canonical bases revisited

Proof. Let e be a hyperimaginary equivalent to some enumeration of D. Then e ∈ dclheq (a) and therefore, by Proposition 15.6, e ∼ aE 0 for some 0type-definable equivalence relation E 0 . By Remark 17.2 it is enough to show Ls that tp(a/e) ` tp(a/bdd(e)). Let a ≡e a 0 . We will check that a ≡e a 0 . By Proposition 16.35, it suffices to check that F (a, a 0 ) for every bounded etype-definable equivalence relation F . Let F be such an equivalence relation. By Lemma 16.31 there is an e-bounded hyperimaginary h such that h ∼ aF e. Since h ∈ bdd(e) and e ∈ bdd(aE ), by Remark 15.19 h ∈ bdd(aE ). Moreover h ∈ dclheq (aF , e) and aF ∈ dclheq (a, e) ⊆ dclheq (a), and hence h ∈ dclheq (a). Therefore h ∈ D ⊆ dclheq (e). If f ∈ Aut(C/e) is such that f(a) = a 0 , then f(h) = h and therefore aF = f(aF ) = f(a)F , that is, F (a, a 0 ). a Definition 17.9. Let e be a hyperimaginary. A canonical base of an amalgamation base p(x) ∈ SE (e) is a smallest hyperimaginary d ∈ dclheq (e) such that p does not fork over d and p  d is an amalgamation base. This means that if d 0 ∈ dclheq (e) is another hyperimaginary such that p does not fork over d 0 and p  d 0 is an amalgamation base then d ∈ dclheq (d 0 ). We will see that in simple theories all amalgamation bases have canonical bases. Remark 17.10. Let T be simple. Assume p(x) ∈ S(A) is stationary. By Lemma 11.2, its global nonforking extension p is definable. Then p is an amalgamation base and two notions of canonical base may be applied to p. It follows from Proposition 17.3 that they are interdefinable. Definition 17.11. Let p(x) be an amalgamation base of hyperimaginary sort. The amalgamation class of p is the class Pp consisting of all global types p(x) such that for some n < ù there are global types (pi (x) : i ≤ n) such that p0 is a nonforking extension of p, p = pn and for every i < n, pi and pi+1 are nonforking extensions of a common amalgamation base pi (x) ⊆ pi , pi+1 . Note that Pp = Pq if and only if Pp ∩ Pq 6= ∅. Remark 17.12. In a simple theory, if p(x) ∈ SE (e) is an amalgamation base over a hyperimaginary e, there is always an amalgamation base q(x) ∈ SE (M ) where M is a model and Pq = Pp . Proof. If p(x) ∈ SE (e) where e is a hyperimaginary, choose a model M such that e ∈ dclheq (M ) and set q(x) = p  M where p is a global nonforking extension of p. a Remark 17.13. Let T be simple and let e be a hyperimaginary. If p(x) ∈ SE (e) is stationary, then the amalgamation class Pp is a singleton {p} where p is the unique global nonforking extension of p. Hence in stable theories amalgamation classes are singletons. Proof. Let p(x) be stationary and let p be its global nonforking extension. If q(x) is an amalgamation base and p, q have a common global nonforking

17. Canonical bases revisited

125

extension, this extension is p and by Proposition 17.3 q(x) is stationary. Hence it is parallel to p and p is its only global nonforking extension. a Lemma 17.14. Let T be simple, let a, b be hyperimaginaries and let p(x) ∈ SE (a) be an amalgamation base. If q(x) ∈ SE (b) is another amalgamation base and Pp = Pq , then for some n < ù there are automorphisms f0 , . . . , fn ∈ Aut(C) such that if pi = p fi , then p0 = p, pi (x) and pi+1 (x) have a common nonforking extension for all i < n, and also pn (x) and q(x) have a common nonforking extension. Proof. It is enough to prove that for all amalgamation bases q(x) ∈ SE (c), r(x) ∈ SE (d ), if p and q have a common nonforking extension and q and r also have a common nonforking extension, then for some automorphism f ∈ Aut(C), p and pf have a common nonforking extension and p f and r also have a common nonforking extension. To check this, let us choose b ≡c a such that b ^ | c ad and let f ∈ Aut(C/c) be such that f(a) = b. Since b ≡c a, pf and q have a common nonforking extension s1 (x) ∈ SE (bc). Now let s2 (x) ∈ SE (cd ) be a common nonforking extension of q and r. Since b ^ | c d and q is an amalgamation base, s1 and s2 have a common extension s(x) ∈ SE (bcd ) which does not fork over c. Then s is a common nonforking extension of r and p f . We finish the proof showing that also p and pf have a common nonforking extension. On the one hand s1 ∈ SE (bc) is a common nonforking extension of p f and q. On the other hand q and p have a common nonforking extension s3 (x) ∈ SE (ac). Since a ^ | c b, we see that s1 and s3 have a common extension s 0 (x) ∈ SE (abc) which does not fork over c. Clearly it is a common nonforking extension of p and p f . a Lemma 17.15. Let T be simple, let R be a 0-type-definable symmetric relation on tuples of a given length. Assume that R is generically transitive, that is, whenever R(a, b), R(a, c), and b ^ | a c, then R(b, c). Let Rn = R ◦ R ◦ · · · ◦ R (n-times) be the n-step composition of R. Then: 1. If R(a, b), then for any tuple d there is some c ≡ b such that R(a, c), R(b, c), bd ^ | a c, and ad ^ | b c. n 2 2. R ⊆ R for all n ≥ 2. 3. If R2 (a, b), then there is some c ≡ b such that R(a, c), R(b, c), b ^ | a c, and a ^ | b c. In particular, since the transitive closure of R is R2 it is 0-type-definable. Proof. 1. Let Σ be the set of all formulas ϕ(x, y) ∈ L with a fixed tuple x of variables and let us define for tuples c of the length of x, ¯ D(c/a) = (D(tp(c/a), ϕ, k) : ϕ ∈ Σ, k ∈ ù).

126

17. Canonical bases revisited

¯ We consider D(c/a) as element of the partially ordered set ù Σ×ù of all sequences (nϕ,k : ϕ ∈ Σ, k ∈ ù). Assume R(a, b) and choose, with Corol¯ lary 3.15, some c with maximal D(c/a) among all tuples c ≡ b such that R(a, c). Without loss of generality, c ^ | a bd . Since R is generically transitive, R(b, c). It is enough to prove that c ^ | b a since from this it follows c ^ | b ad by ordinary computations in the forking calculus. Assume ¯ ¯ ¯ ¯ c^ 6 | b a. Then D(c/b) > D(c/ab). Since c ^ | a b, D(c/ab) = D(c/a). Hence, 0 0 0 ¯ ¯ D(c/b) > D(c/a). Choose c ≡b c such that c ^ | b a. Then R(c , b) and, by generic transitivity, R(c 0 , a). Moreover c 0 ≡ b. Using the fact that c 0 ^ | b a, we 0 0 0 ¯ ¯ ¯ ¯ ¯ see that D(c /a) ≥ D(c /ab) = D(c /b) = D(c/b) > D(c/a), contradicting ¯ the maximality of D(c/a). 2. It is enough to prove that R3 (a, b) implies R2 (a, b). Assume a, a 0 , b 0 , b is an R-path connecting a with b in three steps. Apply 1 to a 0 , b 0 and the additional tuple d = ab. We obtain some c ≡ b 0 such that R(a 0 , c), R(b 0 , c), b 0 ab ^ | a 0 c and a 0 ab ^ | b 0 c. Since R is generically transitive, R(a, c) and R(b, c). 3. Assume R(a, c) and R(b, c). By 1 applied to a, c with d = b we find c 0 ≡ c such that c 0 ^ | a cb and c 0 ^ | c ab. Since R is generically transitive, 0 0 R(b, c ). Now apply 1 to b, c with d = a obtaining c 00 ≡ b such that c 00 ^ | b c 0 a and c 00 ^ | c 0 ba. By generic transitivity of R, R(a, c 00 ). It is easy to 00 check that c ^ | a b and c 00 ^ | b a. a Theorem 17.16. In a simple theory, for every amalgamation base p(x) of hyperimaginary sort there is a hyperimaginary e such that for every automorphism f ∈ Aut(C), f(e) = e if and only if f fixes setwise the amalgamation class Pp . Proof. By Remark 17.12, it is enough to consider an amalgamation base of the form p(x) ∈ SE (a) where a is a real tuple. Write p(x) = p(x, a) with p(x, y) over ∅ and consider the binary relation R on realizations of tp(a) defined by: R(a1 , a2 ) if and only if p(x, a1 ) and p(x, a2 ) have a common nonforking extension. It is symmetric. For each ϕ(x, y) ∈ L, for each k < ù let nϕ,k = D(p(x, a), ϕ, k). Then it is easy to see that R is type-definable by the partial type (over ∅) expressing that a1 , a2 realize tp(a) and for all ϕ ∈ L, for all k < ù, D(p(x, a1 ) ∪ p(x, a2 ), ϕ, k) ≥ nϕ,k . R satisfies all the remaining conditions of Lemma 17.15 and therefore its transitive closure F is also type-definable. Note that F is an equivalence relation on realizations of tp(a) and by Lemma 17.14 F (a, b) holds if and only if Pp(x,b) = Pp(x,a) . By Lemma 15.5 we can extend F to a 0-typedefinable equivalence relation on all sequences of the length of a. Hence we can consider the hyperimaginary e = aF . It is clear that e satisfies the requirements. a Lemma 17.17. Let T be simple, let p(x) ∈ SE (a) be an amalgamation base and let e be a hyperimaginary equivalent to the amalgamation class Pp as in

17. Canonical bases revisited

127

Theorem 17.16. If q(x) ∈ SE (b) is an amalgamation base, Pp = Pq , and a^ | e b then p and q have a common nonforking extension. Proof. By enlarging a, b if necessary, we may assume they are real tuples of the same length. By Lemma 17.14 we may now find n < ù, types pi (x, y) over ∅ and tuples ai for i ≤ n such that p(x) = p0 (x, a0 ), q(x) = pn (x, an ), and pi (x, ai ) and pi+1 (x, ai+1 ) are amalgamation bases with a common nonforking extension for all i < n. We may apply 17.15 to the relation R defined by R(b1 , b2 ) if and only if there are i, j ≤ n such that b1 ≡ ai , b2 ≡ aj and pi (x, b1 ), pj (x, b2 ) have a common nonforking extension. Hence there is an amalgamation base r(x) ∈ SE (c) such that p(x), r(x) have a common nonforking extension, r(x), q(x) have a common nonforking extension, c ^ | a b, and c ^ | b a. Since a ^ | e b and e ∈ dclheq (b) ∩ dclheq (c) from this it follows that a ^ | c b. By amalgamating these types we conclude that p(x) and q(x) have a common nonforking extension. a

Theorem 17.18. Let T be simple, let p(x) ∈ SE (a) be an amalgamation base, let e ∈ dclheq (a) be a hyperimaginary equivalent to the amalgamation class Pp as in Theorem 17.16, and let p0 = p  e. 1. Any p(x) ∈ Pp is a nonforking extension of p0 . 2. p0 is an amalgamation base and Pp = Pp0 . 3. If q(x) ∈ SE (b) is an amalgamation base and p(x), q(x) have a common nonforking extension, then e ∈ dclheq (b). 4. If p(x) and q(x) ∈ SE (b) have a common nonforking extension, then e ∈ bdd(b). 5. If b ∈ dclheq (a), then p(x) does not fork over b if and only if e ∈ bdd(b). Proof. 1. Choose b ≡e a such that b ^ | e a. Let f ∈ Aut(C/e) be such f that f(a) = b and let q(x) = p . Then Pq = Pp and by Lemma 17.17 p(x) and q(x) have a common global nonforking extension. Hence there is a common realization c of p and q such that c ^ | a b. Since e ∈ dclheq (b) by symmetry and transitivity, c ^ | e a and therefore p does not fork over e. Hence any global nonforking extension of p does not fork over e. If p ∈ Pp we can choose an amalgamation base q(x) such that p is a nonforking extension of q(x). Since Pp = Pq , e is a canonical base of q and hence p does not fork over e. 2. Let P = Pp0 . Choose a model M and q(x) ∈ SE (M ) such that e ∈ dclheq (M ) and Pq = P. Then p0 = q(x)  e. Since bdd(e) ⊆ dclheq (M ) and p00 = q(x)  bdd(e) is an amalgamation base, by 1 Pp00 = P. We will see now that all extensions of p0 over bdd(e) have the same property. Let p000 (x) = tp(c/bdd(e)) be any such extension and choose b such that p00 = tp(b/bdd(e)). Then f(b) = c for some f ∈ Aut(C/e). Clearly, f fixes

128

17. Canonical bases revisited

setwise bdd(e) and (p00 )f = p000 . Since f(e) = e, f fixes setwise P, and thus P = Pp000 , as we wanted to show. Next we check that p0 is an amalgamation base. Let M, N be models such that e ∈ dclheq (M ) ∩ dclheq (N ) and M ^ | e N . Assume q1 (x) ∈ SE (M ), q2 (x) ∈ SE (N ) are nonforking extensions of p0 . Note that q1 (x) is a nonforking extension of q1 (x)  bdd(e) and hence Pq1 = P. For similar reasons, Pq2 = P. By Lemma 17.17 q1 (x), q2 (x) have a common nonforking extension which, by symmetry and transitivity, does not fork over e. 3 is clear since Pp = Pq , and 4 follows from 3 because Pp = Pq 0 for some extension q 0 (x) of q(x) over bdd(b). 5. One direction follows from 4. For the opposite direction use 1 to see that p does not fork over e. a Corollary 17.19. Let T be simple and let p(x) ∈ SE (a) be an amalgamation base with amalgamation class Pp . The following are equivalent for any hyperimaginary e: 1. e is a canonical base of p(x). 2. For any f ∈ Aut(C), f(e) = e if and only if f(Pp ) = Pp . Proof. 1 ⇒ 2. Let p be a global nonforking extension of p. If f(e) = e, then p and pf are nonforking extensions of the amalgamation base p  e and therefore pf ∈ Pp and f(Pp ) = Pp . Now assume f(Pp ) = Pp . Let e 0 be a hyperimaginary given as in Theorem 17.18. Then f(e 0 ) = e 0 . Since e 0 ∈ dclheq (a), and p does not fork over e 0 , and p  e 0 is an amalgamation base, we conclude that e ∈ dclheq (e 0 ). Hence f(e) = e. 2 ⇒ 1. By points 1, 2 and 3 of Theorem 17.18. a Corollary 17.20. In a simple theory every amalgamation base has a canonical base. Proof. By Theorem 17.16 and Corollary 17.19.

a

Definition 17.21. If p(x) is an amalgamation base in a simple theory, Cb(p) is, by definition, dclheq (e) where e is a canonical base of p. For any hyperimaginaries a, b we define Cb(a/b) = Cb(tp(a/bdd(b))). As mentioned in Remark 17.10, up to interdefinability this notation agrees with the one introduced for canonical bases of stationary types in simple theories. Remark 17.22. Let T be simple. For all hyperimaginaries a, b: 1. Cb(a/b) ⊆ bdd(b). 2. a ^ | Cb(a/b) b. 3. tp(a/Cb(a/b)) is an amalgamation base. 4. If d ∈ bdd(b), tp(a/d ) is an amalgamation base and a ^ | d b, then Cb(a/b) ⊆ dclheq (d ).

17. Canonical bases revisited

129

Proof. By definition of canonical base, since Cb(a/b) has been defined as Cb(tp(a/bdd(b))). a Lemma 17.23. For simple T the following are equivalent for any hyperimaginaries a, b, c: 1. a ^ | b c. 2. Cb(a/bc) = Cb(a/b). 3. Cb(a/bc) ⊆ bdd(b). Proof. 1 ⇒ 2. If a ^ | b c, then tp(a/bdd(bc)) and tp(a/bdd(b)) have the same amalgamation class. 2 ⇒ 3. Since Cb(a/b) ⊆ bdd(b). 3 ⇒ 1. This follows from the fact that a ^ | Cb(a/bc) bc. a Proposition 17.24. Let T be simple. If p(x) ∈ SE (e) is an amalgamation base over a hyperimaginary e and (ai : i < ù) is a Morley sequence in p, then Cb(p) ⊆ dclheq (ai : i < ù). Moreover, if T is supersimple and p is of real sort, then Cb(p) ⊆ dclheq (ai : i < n) for some n < ù. Proof. Extend the Morley sequence (ai : i < ù) to a Morley sequence (ai : i ≤ ù) in p. By Corollary 17.7 tp(aù /(ai : i < ù)) is an amalgamation base. Since tp(aù /e(ai : i < ù)) is finitely satisfiable in {ai : i < ù}, by Proposition 17.6 aù ^ | (a :i<ù) e. Since aù ^ | e (ai : i < ù) and aù ^ | (a :i<ù) e, i i p = tp(aù /e) and tp(aù /(ai : i < ù)) have the same amalgamation class. Therefore by Theorem 17.18 Cb(p) = Cb(tp(aù /(ai : i < ù)) ⊆ dclheq (ai : i < ù). Assume now T is supersimple and p(x) ∈ S(e) is of real sort. Choose n < ù such that aù ^ | a (ai : i < ù). As before, tp(aù /(ai : i < ù)) is
Chapter 18

ELIMINATION OF HYPERIMAGINARIES

Definition 18.1. T eliminates a hyperimaginary e if there is a sequence of imaginaries (ei : i ∈ I ) such that e ∼ (ei : i ∈ I ). T eliminates hyperimaginaries if T eliminates every hyperimaginary. Proposition 18.2. Let e = aE be a hyperimaginary and let p(x) = tp(a). Then T eliminates e if and only if there is aTfamily (Ei : i ∈ I ) of 0-definable equivalence relations such that E  p = ( i∈I Ei )  p. In fact it suffices to require that the Ei are 0-definable relations whose restrictions Ei  p to p(C) are equivalence relations. Proof. It is enough to require that the Ei are equivalence relations on p(C) since, by compactness, it is always possible to find a formula ϕi (x) ∈ p such that Ei is an equivalence relation on ϕi (C). Assume e = aE is a hyperimaginary and choose a family (Ei : i ∈ I ) of 0-definable equivalenceTrelations such that on p(x) = tp(a) the equivalence relation E agrees with i∈I Ei . Then e ∼ (ei : i ∈ I ) where ei = ai Ei , if ai is the subtuple of a corresponding to the variables of Ei . For the other direction, by assumption there is a sequence (ei : i ∈ I ) of imaginaries ei = ai Ei such that e ∼ (ei : i ∈ I ). Let pi (x, y) = tp(aai ) for each i ∈ I . We may assume the family is closed under finite concatenation in the following sense: for each n < ù, for all i0 , . . . , in ∈ I there is some j ∈ I such that (ei0 , . . . , ein ) ∼ ej . Therefore b1 ≡e b2 if and only if b1 ≡ei b2 for all i ∈ I . Then: 1. E(x, x 0 ) ∪ pi (x, y) ∪ pi (x 0 , y 0 ) ` Ei (y, y 0 ), 2. |= pi (a, ai ), 3. p(x) ` ∃ypi (x, y). By compactness we can substitute a single formula ϕi (x, y) ∈ pi for pi (x, y) and still have property 1. We then define for each i ∈ I , ø(x, y) ∈ pi (x, y): Fi,ø (y, z) ⇔ ∃uv(Ei (u, v) ∧ ϕi (y, u) ∧ ø(y, u) ∧ ϕi (z, v) ∧ ø(z, v)). Clearly Fi,ø is definable over ∅ and Fi,ø  p is an equivalence relation. MoreT over E  p = ( i∈I,ø∈pi Fi,ø )  p. The only point that needs to be checked is that E(b1 , b2 ) if we assume Fi,ø (b1 , b2 ) for all i, ø. It is enough to check it for the case a = b1 . Fix some i ∈ I . By compactness, there are c1 , c2 such 131

132

18. Elimination of hyperimaginaries

that |= pi (a, c1 ), |= pi (b2 , c2 ) and Ei (c1 , c2 ). Since |= pi (a, ai ), Ei (ai , c1 ) and therefore Ei (ai , c2 ). There is some automorphism f such that f(b2 c2 ) = aai . Since f fixes ei , a ≡ei b2 . Hence a ≡e b2 , which implies E(a, b2 ). a Corollary 18.3. T eliminates hyperimaginaries if and only if for any p(x) ∈ S(∅), for any 0-type-definable equivalence relation E on p(C), there T is a family (Ei : i ∈ I ) of 0-definable equivalence relations such that E = ( i∈I Ei )  p. In fact it suffices to require that the Ei are 0-definable relations whose restrictions Ei  p to p(C) are equivalence relations. Proof. By Proposition 18.2. a Lemma 18.4. Let E be an intersection of definable (possibly with parameters) equivalence relations. If E is type-definable over ∅, then E is an intersection of 0-definable equivalence relations. T Proof. Let E = i∈I Ei where every Ei is an equivalence relation, defined by ϕi (x, y, ai ), with ϕi (x, y, z) ∈ L. Assume Σ(x, y) is a type over ∅ defining E and let pi (z) = tp(ai ). Then Σ(x, y) ∪ pi (z) ` ϕi (x, y, z), and therefore Σ(x, y) ` ∀z (øi (z) → ϕi (x, y, z)) for some øi (z) ∈ pi . We can choose it so that øi (z) implies that ϕi (x, y, z) is an equivalence relation in x, y. Then ∀z (øi (z) → ϕi (x, y, z)) defines T (over ∅) an equivalence relation Fi such that E ⊆ Fi ⊆ Ei . Hence E = i∈I Fi . a Proposition 18.5. If T eliminates hyperimaginaries, then T (A) also eliminates hyperimaginaries. Proof. By Lemma 15.20. a Lemma 18.6. If e ∈ dclheq (a) for some sequence of imaginaries a ∈ bdd(e), then e is eliminable. Proof. Let a = (ai : i ∈ I ) where every ai is an imaginary. For each finite J ⊆ I let aJ = (ai : i ∈ J ). Then aJ ∈ aclheq (e) ∩ Ceq . Consider the finite set {f(aJ ) : f ∈ Aut(C/e)} as a single imaginary bJ , as explained in Corollary 1.12. Now let b = (bJ : J ⊆ I is finite ). It is clear that b ∈ dclheq (e). We check now that e ∈ dclheq (b). Assume f fixes b. Then for each finite J , aJ ≡e f(aJ ) and therefore a ≡e f(a). Hence f(a)e ≡ ae ≡ f(a)f(e). Since e ∈ dclheq (a), also f(e) ∈ dclheq (f(a)) and hence e = f(e). a Proposition 18.7. Let T be simple. If e = aE is a hyperimaginary, then e ∈ Cb(a/e). Proof. In a first step we show that e ∈ bdd(Cb(a/e)). By Remark 17.22, a^ | Cb(a/e) e and since e ∈ dclheq (a), we see that e ^ | Cb(a/e) e. By Proposition 16.19, e ∈ bdd(Cb(a/e)). Now we show that e is definable over Cb(a/e). Since tp(a/Cb(a/e)) is an amalgamation base, by Remark 17.2

18. Elimination of hyperimaginaries

133

tp(a/Cb(a/e)) ≡ tp(a/bdd(Cb(a/e))) ` tp(a/e). Let f ∈ Aut(C/Cb(a/e)). Then a ≡Cb(a/e) f(a) and therefore a ≡e f(a). It follows that E(a, f(a)) and hence f(e) = e. a Lemma 18.8. Let T be simple. If a = (ai : i ∈ I ) is a sequence of real elements and for all finite J ⊆ I , aJ = (ai : i ∈ J ), then for every hyperimaginary e, Cb(aJ /e) ⊆ Cb(aJ 0 /e) for J ⊆ J 0 and [ Cb(a/e) = dclheq ( Cb(aJ /e)). J ⊆I finite

Proof. Let C be the union of all Cb(aJ /e) for J ⊆ I finite. Since aJ ^ | Cb(a/e) e and (by Remark 17.4) tp(aJ /Cb(a/e)) is an amalgamation base, by point 4 of remark 17.22, Cb(aJ /e) ⊆ Cb(a/e). Similarly, one can check that Cb(aJ /e) ⊆ Cb(aJ 0 /e) if J ⊆ J 0 . We claim that tp(a/C ) is an amalgamation base. Since a ^ | C e, this will give Cb(a/e) ⊆ dclheq (C ). We use Remark 17.2. By Corollary 16.36 it is enough to show that tp(a/C ) ` tp(a/bdd(Cb(aJ /e)) for all finite J , and this is clear since tp(aJ /Cb(aJ /e)) is an amalgamation base and therefore tp(aJ /Cb(aJ /e)) ` tp(aJ /bdd(Cb(aJ /e))).

a

Proposition 18.9. Let T be simple. If for each finitary amalgamation base p(x), the canonical base Cb(p) is eliminable, then T eliminates hyperimaginaries. Proof. Let e = aE be a hyperimaginary. By Lemma 18.8 and our assumption, there is a sequence d of imaginaries such that Cb(a/e) ∼ d . Then d ∈ bdd(e). By Proposition 18.7 e ∈ Cb(a/e) ⊆ dclheq (d ). By Lemma 18.6 e ∼ d 0 for some sequence of imaginaries d 0 . a Corollary 18.10. Stable theories eliminate hyperimaginaries. Proof. By Proposition 18.9 and by the fact that canonical bases in stable theories are sequences of imaginaries (see Proposition 17.3). a Corollary 18.11. If a supersimple theory eliminates all finitary hyperimaginaries, then it eliminates all hyperimaginaries. Proof. By Proposition 18.9 it suffices to show that T eliminates Cb(p) for all finitary amalgamation bases p(x). This follows from the assumption since by Proposition 17.24 Cb(p) is definable over a finite set and therefore it is finitary. a Proposition 18.12. Let E be a 0-type-definable equivalence relation and let E ∗ be the equivalence relation given by E ∗ (a, b) ⇔ E(a 0 , b 0 ) for some a 0 ≡ a, b 0 ≡ b.

134

18. Elimination of hyperimaginaries

Then E is an intersection of 0-definable equivalence relations if and only if E ∗ is an intersection of 0-definable equivalence relations and for each p(x) ∈ S(∅), E  p = E ∩(p(C)×p(C)) is an intersection of 0-definable equivalence relations. Proof. Note that each of the two following conditions is equivalent to E ∗ (a, b): 1. E(a, c) for some c ≡ b. 2. E(a, a 0 ) and E(b, b 0 ) for some a 0 ≡ b 0 . If E ∗ (a, b) is witnessed by E(a 0 , b 0 ) where a 0 ≡ a and b 0 ≡ b, and we choose c such that ac ≡ a 0 b 0 then c witnesses that 1 holds. For 1 ⇒ 2 just take a 0 = c and b 0 = b. Finally if a 0 , b 0 are as in 2 and we choose c such that a 0 a ≡ b 0 c, then E(c, b), a ≡ c and T b ≡ b. Now assume E = i∈I Ei for 0-definable equivalence relations Ei . Then T obviously for each p ∈ S(∅), E agrees with i∈I Ei on p. Moreover E ∗ = T i∈I,ϕ∈L Eiϕ where Eiϕ (x, y) is the equivalence relation defined by ∃z(ϕ(z) ∧ Ei (x, z)) ↔ ∃z(ϕ(z) ∧ Ei (y, z)). For the other direction, suppose that E ∗ is an intersection T of 0-definable equivalence relations and for each p(x) ∈ S(∅), E  p = i∈Ip Eip  p for a family of 0-definable equivalence relations Eip . We can assume that the type E(x, y) defining the equivalence relation E is made of symmetric formulas. Fix some p(x) ∈ S(∅) and choose some a |= p. For each i ∈ Ip we can find some formula óip (x, y) ∈ E(x, y) and some øip (x) ∈ p such that óip (x, y) ∧ øip (x) ∧ øip (y) ` Eip (x, y). We can also find some ó ip (x, y) ∈ E(x, y) such that ó ip (x, y) ∧ ó ip (y, z) ∧ ó ip (z, u) ` óip (x, u) ∗ and some 0-definable equivalence relation Eip in the family whose intersection ∗ is E such that ∗ Eip (x, a) ` ∃y(øip (y) ∧ ó ip (x, y)). ∗ Consider the relation Fip (x, y) defined by the disjunction of (¬Eip (x, a) ∧ ∗ ¬Eip (y, a)) with ∗ ∗ (Eip (x, a)∧Eip (y, a)∧∃uv(øip (u)∧øip (v)∧ó ip (x, u)∧ó ip (y, v)∧Eip (u, v))).

Note that the definition is in fact independent of the choice of the realization a of p. It is clearly reflexive and symmetric. It is not difficult to see that it is also transitive. We claim that \ E = E∗ ∩ Fip . p∈S(∅),i∈Ip

By Lemma 18.4 this will show that E is an intersection of 0-definable equivalence relations.

18. Elimination of hyperimaginaries

135

Assume E(c, d ). Then E ∗ (c, d ). Let p(x) ∈ S(∅), let i ∈ Ip , and let a |= p. ∗ ∗ We want to check that Fip (c, d ). We may assume Eip (c, a) ∧ Eip (d, a). By ∗ 0 0 0 choice of Eip we know that there are c , d such that |= øip (c ) ∧ ó ip (c, c 0 ) and |= øip (d 0 ) ∧ ó ip (d, d 0 ). Then |= óip (c 0 , d 0 ) and therefore Eip (c 0 , d 0 ). For the other direction, assume E ∗ (c, d ) and Fip (c, d ) for all p, i. Let p(x) = tp(c). As remarked above, E(c 0 , d ) for some c 0 ≡ c. It is enough to show that E(c, c 0 ) and for this we have to check that Eip (c, c 0 ) for all i ∈ Ip . Note that Fip (c, d ) and Fip (d, c 0 ) since we have already shown that E(x, y) implies Fip (x, y). Hence Fip (c, c 0 ) and by definition of Fip there are b, b 0 such that |= øip (b) ∧ ó ip (c, b) ∧ øip (b 0 ) ∧ ó ip (c 0 , b 0 ) ∧ Eip (b, b 0 ). Note that |= øip (c) ∧ øip (b) ∧ óip (c, b) and thus Eip (c, b). Similarly Eip (c 0 , b 0 ) and we conclude Eip (c, c 0 ). a Lemma 18.13. Let T be small and let E be a 0-type-definable equivalence relation on Cn such that: if E(a, b), a ≡ a 0 and b ≡ b 0 , then E(a 0 , b 0 ). Then E is an intersection of 0-definable equivalence relations. Proof. We claim that whenever ¬E(a, b), then for some ϕab ∈ L, |= ϕab (a) ∧ ¬ϕab (b) and E(x, y) ∧ ϕab (x) ∧ ¬ϕab (y) is inconsistent. If this is the case, we can then express E as an intersection of 0-definable equivalence relations as follows: ^ E(x, y) ⇔ ϕab (x) ↔ ϕab (y). ¬E(a,b)

In order to prove this claim, assume ¬E(a, b) and set p(x) = tp(a), q(x) = tp(b). Recall that, by Remark 14.27, all types in Sn (∅) have ordinal Cantor– Bendixson rank. We first observe that E(x, y)∪p(x)∪q(y) is inconsistent and hence we can choose ϕ(x) ∈ p(x), ø(y) ∈ q(y) such that E(x, y) ∧ ϕ(x) ∧ ø(y) is inconsistent and ¬ϕ(x)∧¬ø(x) is of minimal Cantor–Bendixson rank α in the space Sn (∅) and of minimal degree in this rank. If ¬ϕ(x) ∧ ¬ø(x) is inconsistent we set ϕab = ϕ and this choice satisfies the requirements. Otherwise we choose a type p 0 (x) ∈ Sn (∅) of rank α containing the formula ¬ϕ(x) ∧ ¬ø(x) and also a realization c |= p 0 . Now, if there if some a 0 |= ϕ and some b 0 |= ø such that E(a 0 , c) and E(b 0 , c) then E(x, y) ∧ ϕ(x) ∧ ø(y) turns out to be consistent. Hence we may assume that there is no a 0 |= ϕ such that E(a 0 , c), that is, E(x, y) ∧ ϕ(x) ∧ p 0 (y) is inconsistent. Therefore E(x, y) ∧ ϕ(x) ∧ ø 0 (y) is inconsistent for some ø 0 ∈ p 0 . Note that either ¬ϕ(x) ∧ ¬ø 0 (x) has rank < α or has rank α and smaller degree than ¬ϕ(x) ∧ ¬ø(x). This contradicts the previous choice of ϕ(x) and ø(x). a Theorem 18.14. Let T be small. 1. If E is a 0-type-definable equivalence relation on Cn , then E is an intersection of 0-definable equivalence relations. 2. T eliminates all finitary hyperimaginaries. KP s 3. For any finite set A, ≡A = ≡A .

136

18. Elimination of hyperimaginaries Ls

s

4. If T is G-compact over A, in particular if T is simple, then ≡A = ≡A . Proof. 1. We apply Proposition 18.12. It is clear that E ∗ satisfies the hypothesis of Lemma 18.13 and therefore it is an intersection of 0-definable equivalence relations. Now fix some p(x) ∈ Sn (∅) and choose some c |= p. We have to show that T for some family (Ei : i ∈ I ) of 0-definable equivalence relations, E  p = i∈I Ei  p. Consider the relation F (x, y) ⇔ for some z, cx ≡ zy and E(c, z). It is an equivalence relation and it is type-definable over c. Since T (c) is small and in T (c) the relation F satisfies the hypothesis of Lemma 18.13, T there is some family (Fi : i ∈ I ) of equivalence relations Fi such that F = i∈I Fi and for each i ∈ I there is some ϕi (x, y, z) ∈ L such that ϕi (x, y, c) defines Fi . Now let Ei (x, y) ⇔ ∀u(ϕi (u, x, x) ↔ ϕi (u, y, y)). T It is clearly an equivalence relation. We check that E  p = i∈I Ei  p. It suffices to see that for any a |= p, E(a, c) if and only if Ei (a, c) for all i ∈ I . Assume E(a, c), let i ∈ I , let b be arbitrary and choose b 0 such that ab ≡ cb 0 . Then F (b, b 0 ) and therefore |= ϕi (b, b 0 , c). Since ϕi (x, y, c) defines an equivalence relation, |= ϕi (b, c, c) ↔ ϕi (b 0 , c, c). By automorphism, |= ϕi (b, c, c) ↔ ϕi (b, a, a) and thus Ei (a, c). For the other direction, assume Ei (a, c) for all i ∈ I . Since |= ϕi (a, a, a), we obtain |= ϕi (a, c, c) and hence F (a, c). This clearly implies E(a, c). 2. It follows from 1 and Corollary 18.3. 3. Since T (A) is again small we may assume A = ∅. It is enough to check the equality for finite sequences and this case follows straightforwardly from KP 1 since it implies that on n-tuples ≡ is an intersection of finite 0-definable equivalence relations. 4. If T is simple, then by Corollary 10.17 T is G-compact over A, that is, KP Ls KP KP ≡A = ≡A . By Remark 15.22, a ≡A b if and only if a ≡A0 b for all finite s A0 ⊆ A. The same for ≡A . Then we can apply 3. a Example 18.15. 1. (Pillay–Poizat in [35]) There is a superstable theory T (of U-rank 1) where we can find a 0-type-definable equivalence relation which is not an intersection of 0-definable equivalence relations. 2. (Adler in [1]) There is an ù-categorical (hence small) theory which does not eliminate hyperimaginaries. More generally, in any theory with the strict order property there is an infinitary hyperimaginary which is not eliminable. Definition 18.16. A formula ϕ(x, y) ∈ L is low if there is some n < ù such that for any indiscernible sequence (ai : i < ù), if {ϕ(x, ai ) : i < ù} is inconsistent, then it is n-inconsistent. We say that T is low if it is simple and every formula is low in T .

18. Elimination of hyperimaginaries

137

Definition 18.17. Let ϕ(x, y) ∈ L. For any set of formulas ð(x) we define the rank D(ð, ϕ) as follows 1. D(ð, ϕ) ≥ 0 if and only if ð(x) is consistent. 2. D(ð, ϕ) ≥ α + 1 if and only if for some a, ϕ(x, a) divides over the parameters of ð and D(ð ∪ {ϕ(x, a)}, ϕ) ≥ α. 3. D(ð, ϕ) ≥ α if and only if D(ð, ϕ) ≥ â for all â < α if α is a limit ordinal number. Remark 18.18. 1. D(ð, ϕ, k) ≤ D(ð, ϕ) ≤ D(ð). 2. If ð(x) is a partial type over A, then D(ð, ϕ) ≥ α + 1 if and only if for some a, ϕ(x, a) divides over A and D(ð ∪ {ϕ(x, a)}, ϕ) ≥ α. Proposition 18.19. Let T be simple and let ϕ(x, y) ∈ L. The following are equivalent: 1. 2. 3. 4. 5. 6.

ϕ(x, y) is low. There is some k < ù such that for all ð, D(ð, ϕ) = D(ð, ϕ, k). D(x = x, ϕ) < ù. There is some n < ù such that for all k < ù, D(x = x, ϕ, k) < n. There is some n < ù such that ϕ divides at most n times. {(a, b) ∈ C : ϕ(x, a) divides over b} is type-definable over ∅ ( for any fixed length of b).

Proof. 1 ⇒ 2. Fix n < ù as in the definition of low. If ϕ(x, a) divides over A, it divides over A with respect to n. Hence D(ð, ϕ) = D(ð, ϕ, n). 2 ⇒ 3. By simplicity, D(x = x, ϕ, k) < ù. 3 ⇒ 4 is clear since D(x = x, ϕ, k) ≤ D(x = x, ϕ). 4 ⇒ 5. Fix n as in 4. If ϕ divide m times, there are tuples (ai : i < m) and numbers (ki : i < m) such that {ϕ(x, ai ) : i < m} is consistent and for each i < m, ϕ(x, ai ) divides over a
138

18. Elimination of hyperimaginaries

Assume ð(x, y) is a partial type over ∅ defining dividing as in 6. Since, for each k < ù, ϕ(x, aùk ) divides over bk = (aik : i < ù), we have |= ð(aùk , bk ) and therefore |= ð(cù , c). But then ϕ(x, cù ) divides over c. a Remark 18.20. Conditions 3, 4, 5 of Proposition 18.19 are equivalent in any theory T . Moreover, if they hold for any ϕ, the theory T is simple. Proof. Note that 5 ⇒ 3 follows from Proposition 3.8. a Remark 18.21. If T is low, then T (A) is also low for any set A. Proof. This is clear, for instance, from point 6 of Proposition 18.19 since ϕ(x, a, b, c) divides over bc in T if and only if ϕ(x, a, b, c) divides over b in T (c). a Proposition 18.22. 1. Any stable theory is low. 2. Any supersimple theory of finite D-rank is low. Proof. By Proposition 18.19, 2 is clear, since D(x = x, ϕ) ≤ D(x = x). For 1, assume {ϕ(x, ai ) : i < n} V is consistent and ϕ(x, ai ) divides over (aj : j < i) for each i < n. Let b |= i CBϕ (pi+1 ) for all i < n and therefore ù > CBϕ (x = x) ≥ D(x = x, ϕ). a Remark 18.23. There are supersimple nonlow theories. An example is given in [9]. Definition 18.24. There is a natural topology in CI , the topology whose closed sets are the type-definable (with parameters) subsets of CI . By analogy with the case of an algebraically closed field, let us call it the Zariski topology. If E is a 0-type-definable equivalence relation in a type-definable over ∅ subclass of CI , the logic topology or the Kim–Pillay topology is the quotient topology of the Zariski topology. If ð(x) is the type defining the domain of E and X = ð(C)/E is the quotient, then A ⊆ X is closed if and only if {a |= ð : aE ∈ A} is type-definable. As usual, in this context we will always identify E with the type defining it and we will assume that the type E(x, y) is closed under finite conjunctions and that E(x, y) ` ð(x) ∪ ð(y). Proposition 18.25. Let ð(x) be a type over ∅, let E be a 0-type-definable equivalence relation on ð(C), and let us work in the Kim–Pillay space X = ð(C)/E. 1. Y ⊆ X is closed if and only if for some type-definable class A, Y = {aE : a ∈ A}. 2. X is Hausdorff. 3. A basis of open sets is given by the collection of all Uaϕ = {bE : |= ϕ(a 0 , b 0 ) for all a 0 , b 0 such that E(a, a 0 ), E(b, b 0 )} where a |= ð and ϕ = ϕ(x, y) ∈ E. 4. X is compact if and only if E is bounded.

18. Elimination of hyperimaginaries

139

Proof. 1. If A = Φ(C), then {a : aE ∈ Y } is defined by the type Ψ(x) = ∃y(E(x, y) ∧ Φ(y)). 2 is clear. We check 3. Note that {b |= ð : bE 6∈ Uaϕ } is type-definable and hence Uaϕ is open. Let U be open and aE ∈ U . Choose a partial type Σ(x) (extending ð) such that X r U is the set of all bE such that b |= Σ. Choose ó ∈ Σ such that |= ¬ó(a). Note that E(x, y) ∧ Σ(x) ` Σ(y) and hence ϕ(x, y) ∧ Σ(x) ` ó(y) for some ϕ(x, y) ∈ E(x, y). Then aE ∈ Uaϕ ⊆ U . 4. Assume first E is bounded and let (Fi : i ∈ I ) be a family of closed sets with the finite intersection property. For each i ∈ I choose a type Φi (x) such that Fi = {aE : a |= Φi }. If the number of E-classes is bounded by κ κ κ, the number S of closed sets in X is bounded by 2 and hence |I | ≤ 2 . Therefore i∈I Φi is a partial type over a subset of C and we can realize it by some a ∈ C. Clearly, aE ∈ Fi for all i ∈ I . For the other direction, assume now X is compact. Fix ϕ(x, y) ∈ E(x, y). We will show that ϕ is finite on ð, that is, there is no infinite sequence (ai : i ∈ ù) of realizations ai of ð such that |= ¬ϕ(ai , aj ) for all i < j < ù. From this it follows that E is bounded. Assume there is such a sequence (ai : i ∈ ù). We can extend it to a maximal one (ai : i ∈ IS ). Then for any a |= ð there is some i ∈ I such that |= ϕ(a, aS i ), that is X ⊆ i∈I Uai ϕ . By compactness of X , for some finite I0 ⊆ I , X ⊆ i∈I0 Uai ϕ . This contradicts the choice of the sequence. a Proposition 18.26. Let ð(x) be a type over ∅, let E be a 0-type-definable equivalence relation on ð(C), and consider the Kim–Pillay space X = ð(C)/E. The following conditions are equivalent. 1. X is 0-dimensional. 2. E is an intersection of 0-definable equivalence relations. 3. For each ϕ(x, y) ∈ E there is some ϕ 0 (x, y) implied by E(x, y) and such that (a) ð(x) ∪ ð(y) ` ϕ 0 (x, y) → ϕ(x, y), (b) E(x, x 0 ) ∪ E(y, y 0 ) ` ϕ 0 (x, y) → ϕ 0 (x 0 , y 0 ). Proof. 1 ⇒ 2. Let (Oi : i ∈ I ) be a basis of clopen sets. For each i ∈ I there is some formula ϕi (x) ∈ L(C) such that {a |= ð : |= ϕi (a)} = {a |= ð : aE ∈ Oi }. T Let a |= ð. Since {aE } is closed, there is a subset Ia ⊆ I such that {aE } = i∈Ia Oi . For each i ∈ I , (ϕi (x) ↔ ϕi (y)) defines an equivalence relation. It is easy to check that E can be defined by ^ ^ (ϕi (x) ↔ ϕi (y)). a|=ð i∈Ia

T

2 ⇒ 3. Let E = i∈I Ei where each Ei is a 0-definable equivalence relation. If ϕ(x, y) ∈ E(x, y), then for some i ∈ I , Ei (x, y) ` ϕ(x, y) and clearly ϕ 0 (x, y) = Ei (x, y) satisfies all the requirements.

140

18. Elimination of hyperimaginaries

3 ⇒ 1. Let (Uaϕ : a |= ð, ϕ ∈ E) be the basis of open sets described in Proposition 18.25. For each ϕ ∈ E choose ϕ 0 as in 3. Then (Uaϕ 0 : a |= ð, ϕ ∈ E) is again a basis of open sets. It is easy to check that in fact each Uaϕ 0 is clopen. a Proposition 18.27. If T is G-compact over ∅ (in particular, if T is simple), then T eliminates all bounded hyperimaginaries if and only if Lstp = stp, that Ls s is, if and only if for all sequences a, b: a ≡ b if and only if a ≡ b. Proof. G-compactness over ∅ means Autf(C) = Aut(C/bdd(∅)) and also KP Ls means ≡ = ≡. It is clear that if T eliminates all bounded hyperimaginaries, then Aut(C/bdd(∅)) = Aut(C/acleq (∅)) and therefore Lstp = stp. For the other direction, let e = aE be a bounded hyperimaginary. By Proposition 15.27 we can assume E is a bounded equivalence relation. By GLs compactness, ≡ is the least bounded 0-type-definable equivalence relation and therefore aE splits into a bounded number of Lascar strong types. By assumption and by Proposition 18.2 for each bEa there is a sequence of imaginaries b 0 such that b Ls ∼ b 0 . Let (bi : i ∈ I ) be a sequence of representatives ≡

of Lascar strong types of elements in aE . Then e ∈ dclheq (bi0 : i ∈ I ) and (bi0 : i ∈ I ) ∈ bdd(e). By Lemma 18.6 e is equivalent to a sequence of imaginaries. a Lemma 18.28. Assume T is a simple theory. Let p(y) ∈ S(A) and let ø1 (x, y), . . . , øn (x, y) ∈ L(A). Then {(a1 , . . . , an ) : a1 , . . . , an are A-independent realizations of p and the formula ø1 (x, a1 ) ∧ · · · ∧ øn (x, an ) does not fork over A} is type-definable over A. Proof. By Proposition 5.15 and Corollary 5.24, noticing that if each bi = (ai1 , . . . , ain ) is an A-independent sequence of realizations aij of p, then (bi : i < ù) is a Morley sequence over A if and only if the composed sequence (ck : k < ù) (where ci·n+j = aij ) is a Morley sequence over A. a Theorem 18.29. Let T be a low theory. 1. T eliminates all bounded hyperimaginaries. s Ls 2. For any set A, for all tuples a, b: a ≡A b if and only if a ≡A b. Proof. 1 follows from 2 and Proposition 18.27. Ls 2. By Remark 18.21, we can assume A = ∅. Let E =≡ and consider e = aE , a bounded hyperimaginary. We will show that we can eliminate e using Proposition 18.2. Let p(x) = tp(a). We use point 3 of Proposition 18.26 to show that E  p is an intersection of 0-definable equivalence relations. Let ϕ(x, y) ∈ E  p. We need to find ϕ 0 (x, y) ∈ E  p such that ϕ 0 (x, y) ` ϕ(x, y) and E(x, x 0 ) ∧ E(y, y 0 ) ∧ ϕ 0 (x, y) ` ϕ 0 (x 0 , y 0 ). Choose ϕ¯ ∈ E(x, y) 

18. Elimination of hyperimaginaries

141

p such that ϕ(x, ¯ y) ∧ ϕ(y, ¯ z) ∧ ϕ(z, ¯ u) ∧ ϕ(u, ¯ v) ` ϕ(x, v). Consider the following binary relation R(b, c) on realizations b, c of p: R(b, c) ⇔ ϕ(x, ¯ b 0 ) ∧ ϕ(x, ¯ c 0 ) does not fork over ∅ for some b 0 , c 0 |= p such that E(b, b 0 ), E(c, c 0 ), and b 0 ^ | c 0. We will check that R is definable by some formula ϕ 0 as above. Since e ∈ bdd(∅), a ^ | e and hence the type E(x, a) does not fork over ∅. Likewise, for any b |= p the type E(x, b) does not fork over ∅. This implies that we can find an independent sequence b1 , b2 , b3 in the E-class bE , which shows that ϕ(x, ¯ b1 ) ∧ ϕ(x, ¯ b2 ) does not fork over ∅. It follows that whenever b, c |= p and E(b, c) then R(b, c). By choice of ϕ, ¯ whenever R(b, c) then |= ϕ(b, c). Finally, it is obvious that if E(b, b 0 ), E(c, c 0 ), and R(b, c), then R(b 0 , c 0 ). To check the definability of R we show that R and its complement are type-definable. Type-definability of R follows from Lemma 18.28. For the complement R¯ of R we need to use the lowness of T . First note that, since Ls E = ≡, using Corollary 10.8 it is easy to see that for all b, c, b 0 , c 0 realizing p, if b ^ | c, b 0 ^ | c 0 , E(b, b 0 ), and E(c, c 0 ), if ϕ(x, ¯ b) ∧ ϕ(x, ¯ c) does not fork over ∅, then also ϕ(x, ¯ b 0 ) ∧ ϕ(x, ¯ c 0 ) does not fork over ∅. Hence for b, c |= p, ¯ c) if and only if there are b 0 , c 0 |= p such that E(b, b 0 ), E(c, c 0 ), b 0 | c 0 R(b, ^ and ϕ(x, ¯ b 0 ) ∧ ϕ(x, ¯ c 0 ) forks over ∅. By Proposition 18.19 and Corollary 5.24 it is easily seen that this relation is type-definable over ∅. a

Chapter 19

ORTHOGONALITY AND ANALYSABILITY

Definition 19.1. Let p(x) ∈ SE (e), q(y) ∈ SF (d ) where e, d are hyperimaginaries. If e, d ∈ dcleq (h), we say that p, q are orthogonal over h and we write p ⊥h q if , if whenever p0 (x) ∈ SE (h) is a nonforking extension of p and q 0 (y) ∈ SF (h) is a nonforking extension of q, then a ^ | h b for all a |= p0 , for all b |= q 0 . We say that p, q are orthogonal and we write p ⊥ q if p ⊥h q for every hyperimaginary h over which e, d are definable. Lemma 19.2. Let T be simple, let e, d be hyperimaginaries and let p(x) ∈ SE (e), q(y) ∈ SF (d ). If for every hyperimaginary h such that e, d ∈ dclheq (h) there is some set C such that h ∈ dclheq (C ) and p ⊥C q, then p ⊥ q. Proof. Let e, d ∈ dclheq (h) and let h ∈ dclheq (C ) be such that p ⊥C q. Let p (x), q 0 (x) be nonforking extensions of p, q over h and let a |= p 0 , b |= q 0 . Let a 0 b 0 ≡h ab be such that a 0 b 0 ^ | h C . Then a 0 , b 0 realize nonforking extensions of p, q over C and, by assumption, a 0 ^ | C b 0 . It follows that 0 0 a ^ | h b and therefore a ^ | h b. a 0

Proposition 19.3. Let T be simple, let e, d be hyperimaginaries and let p(x) ∈ SE (e), q(y) ∈ SF (d ). Let e 0 and d 0 be hyperimaginaries such that e ∈ dclheq (e 0 ) and d ∈ dclheq (d 0 ). Then p ⊥ q if and only if p 0 ⊥ q 0 for all nonforking extensions p 0 ∈ SE (e 0 ) and q 0 ∈ SF (d 0 ) of p, q respectively. Proof. From left to right it is clear. Use Lemma 19.2 for the opposite direction. a Lemma 19.4. Let T be simple and let e, d be hyperimaginaries. If tp(a/e) is orthogonal to p(x) ∈ SE (d ) and a 0 ∈ bdd(ae), then tp(a 0 /e) ⊥ p. a

Proof. Easy to check.

Proposition 19.5. Let T be simple and let e, b be hyperimaginaries. Assume a = (ai : i ∈ I ) is a sequence of hyperimaginaries and it is independent over e. If tp(ai /e) ⊥ tp(b/e) for all i ∈ I , then tp(a/e) ⊥ tp(b/e). Proof. Assume that d is a hyperimaginary such that e ∈ dclheq (d ) and let a = (ai0 : i ∈ I ) ≡e a and a 0 ^ | e d and let b 0 ≡e b and b 0 ^ | e d . Then a 0 is 0

143

144

19. Orthogonality and analysability

independent over d . By induction on n < ù it is easy to check that whenever i1 , . . . , i n ∈ I , b 0 ^ | d ai01 . . . ai0n , which implies b 0 ^ | d a0. a Lemma 19.6. Let T be simple and let A be a set. Assume (ai : 1 ≤ i ≤ m) is A-independent, and each tp(ai /A) is orthogonal to all types of D-rank < α. If D(tp(dj /A)) ≤ α for all j = 1, . . . , n, and d1 , . . . , dn ^ 6 | A ai for all i = 1, . . . , m, then m ≤ n. Proof. The proof is by induction on n. We use Remark 14.6 several times. The starting case is n = 1. Assume d1 ^ 6 | A a1 and d1 ^ 6 | A a2 . Then D(tp(d1 /Aa2 )) < α and therefore tp(a1 /A) ⊥ tp(d1 /Aa2 ). Since a1 ^ | A a2 , by orthogonality a1 ^ | Aa d1 and therefore a1 ^ | A d1 , a contradiction. 2 Now consider the case n + 1. Assume d1 , . . . , dn+1 ^ 6 | A ai for all i ≤ m, and assume m > n + 1. We claim that dj ^ 6 | A a1 , . . . , an+1 for all j ≤ n + 1. Assume dj ^ | A a1 , . . . , an+1 for some j ≤ n + 1, say for j = n + 1. Then d1 , . . . , d n ^ 6 | Ad ai for all i ≤ n + 1. This contradicts the induction hypothen+1

sis, since a1 , . . . , an+1 are independent over Adn+1 , tp(ai /Adn+1 ) is orthogonal to every type of D-rank < α for all i ≤ n, and D(tp(dj /Adn+1 )) ≤ α for all j ≤ n. Now we claim that an+2 ^ | A a1 , . . . , an+1 d1 , . . . , dj for all j ≤ n + 1, a contradiction with the assumption for the case j = n + 1. We prove it by induction on j. Consider the case j = 1. Since D(tp(d1 /Aa1 , . . . , an+1 )) < α and an+2 ^ | A a1 , . . . , an+1 , by orthogonality an+2 ^ | Aa ,...,a d1 . It fol1 n+1 lows that an+2 ^ | A a1 , . . . , an+1 d1 . Now consider the case j + 1, assuming an+2 ^ | A a1 , . . . , an+1 d1 , . . . , dj . Since dj+1 ^ 6 | A a1 , . . . , an+1 d1 , . . . , dj , it follows that D(dj+1 /Aa1 , . . . , an+1 d1 , . . . , dj )) < α. By orthogonality we obtain that an+2 ^ | Aa ,...,a d ,...,d dj+1 . By transitivity we conclude that an+2 ^ | A a1 , j 1 n+1 1 . . . , an+1 d1 , . . . , dj+1 . a Definition 19.7. Let P be a family (possibly a proper class) of partial types and let e be a hyperimaginary. Assume that P is e-invariant, that is, for every f ∈ Aut(C/e), for every ð ∈ P, ðf ∈ P. A hyperimaginary type p(x) ∈ SE (e) is internal in P if for every hyperimaginary a |= p there is some set B such that e ∈ dclheq (B), and for some some sequence (ði : i ∈ I ) of partial types ði ∈ P over B there is a sequence c = (ci : i ∈ I ) of realizations ci |= ði such that a^ | B and a ∈ dclheq (B, c). e

The type p(x) ∈ SE (e) is foreign to P if for every hyperimaginary a |= p, for every set A such that e ∈ dclheq (A) and a ^ | e A, for every tuple c of realizations of types in P over A, a ^ | A c. Notice that in both cases it suffices to find some realization a of p with the corresponding properties.

19. Orthogonality and analysability

145

Lemma 19.8. Let T be simple, and let e, d be hyperimaginaries. Let p(x) ∈ SE (e), let e ∈ dclheq (d ), and let q(x) ∈ SE (d ) be a nonforking extension of p. Let P be an e-invariant family of partial types. 1. p is internal in P if and only if q is internal in P. 2. If p is foreign to P, then p 0 is foreign to P. 3. If p is an amalgamation base and p 0 is foreign to P, then p is foreign to P. Proof. Easy exercise. a Remark 19.9. Let T be simple let a, e be hyperimaginaries and let P be an e-invariant family of partial types. If tp(a/e) is internal in P and foreign to P, then a ∈ bdd(e). Proof. Since tp(a/e) is internal in P, there is some set A such that e ∈ dclheq (A) and a ^ | e A, and there is some tuple c of realizations of types in P over A such that a ∈ dclheq (A, c). Since tp(a/e) is foreign to P, a ^ | A c and therefore a ^ | A a. Then a ^ | e a and hence a ∈ bdd(e). a Lemma 19.10. Let T be simple, let a, b, e be hyperimaginaries, and let P be an e-invariant family of partial types. 1. If tp(a/e) is internal in P and b ∈ dclheq (ae), then tp(b/e) is internal in P. 2. If tp(a/e) is internal in P, then tp(a/eb) is internal in P. Proof. Easy to check. a Lemma 19.11. Let T be simple, let a, b, e be hyperimaginaries and assume P is an e-invariant family of partial types and tp(a/e) is internal in P. If a 0 = Cb(a/eb) then tp(a 0 /e) is internal in P. Proof. Choose a set A and a tuple c of realizations of types in P over A such that e ∈ dclheq (A), a ^ | e A and a ∈ dclheq (A, c). Let (ai : i < ù) be a Morley sequence in tp(a/be). By Proposition 17.24 a 0 ∈ dclheq (ai : i < ù). For each i < ù choose ci , Ai such that acA ≡e ai ci Ai . Then ci is a tuple of realizations of types in P over Ai , e ∈ dclheq (Ai ), ai ^ | e Ai and ai ∈ dclheq (Ai , ci ). If necessary, we replace each Ai by A0i ≡eai Ai such that A0i ^ | e A0
a 0 ∈ dclheq (A, (ci : i < ù)), tp(a 0 /e) is internal in P. a Lemma 19.12. Let T be simple, let a, e be hyperimaginaries and assume P is an e-invariant family of partial types. If there is some b ∈ bdd(ae) r bdd(e) such that tp(b/e) is internal in P, then there is some d ∈ dclheq (ae) r bdd(e) such that tp(d/e) is internal in P. Proof. Consider the orbit X = {f(b) : f ∈ Aut(C/ae)}. By Proposition 15.28, there is a hyperimaginary d such that for every automorphism f ∈ Aut(C), f(d ) = d if and only if f(X ) = X . Fix an enumeration

146

19. Orthogonality and analysability

X = {bi : i < α} for some ordinal α. Each tp(bi /e) is internal in P and hence we can find some set Bi such that e ∈ dclheq (Bi ) and bi ^ | e Bi and there is

some tuple ci of realizations of types in P over Bi such that bi ∈ dclheq (Bi , ci ). 0 Choose inductively Bi0 ≡ebi Bi such that Bi0 ^ | eb (bj : j < α)B
Notation 19.14. For any ordinal α, let Pα be the family of all formulas of D-rank α. Notice that Pα is invariant S and hence it is e-invariant for all hyperimaginaries e. We also put P≤α = i≤α Pi . Corollary 19.15. Let T be simple. Let e, a be hyperimaginaries. If α is the minimal ordinal such that tp(a/e) 6⊥ p for some type p of D-rank α, then there is some hyperimaginary d ∈ dclheq (ae) r bdd(e) such that tp(d/e) is internal in the family Pα of all formulas of D-rank α. Proof. Since any extension of p has D-rank ≤ α, we may assume p is a type over a set of real parameters A such that e ∈ dclheq (A). Let P be the family of all e-conjugates of p. Now, tp(a/e) is nonorthogonal to any type in P and therefore it is not foreign to P. By Proposition 19.13, there is some

19. Orthogonality and analysability

147

d ∈ dclheq (ae) r bdd(e) such that tp(d/e) is internal in P. Hence tp(d/e) is also internal in Pα . a Definition 19.16. Let e be a hyperimaginary and let P be an e-invariant family (as above, possibly a proper class) of partial types. A hyperimaginary type p(x) ∈ SE (e) is analysable in P if for every hyperimaginary a |= p there is some ordinal α and some sequence (ai : i < α) ∈ dclheq (a, e) of hyperimaginaries ai such that tp(ai /ea 0 and the result holds for all e 0 such that SU(a/e 0 ) < α. There is a finite tuple b such that a ^ 6 | e b. Then tp(a/e) 6⊥ tp(b/e) and (see Proposition 19.3) tp(a/e) 6⊥ tp(b/A) where A is a set of real elements such that e ∈ dclheq (A) and b ^ | e A. Since T is supersimple and b is finite, D(tp(b/A)) < ∞. Let α0 be the least ordinal such that tp(a/e) is nonorthogonal to a type p(x) of D-rank α0 . By Corollary 19.15 there is some hyperimaginary d ∈ dclheq (ae) r bdd(e) such that tp(d/e) is internal in Pα0 . By minimality of α0 , tp(a/e) is orthogonal to all types of D-rank < α0 . Let a1 = de and a0 = e. By Lemma 19.10 tp(a1 /a0 ) is internal in Pα0 . Since d ∈ dclheq (ae) r bdd(e), a ^ 6 | e d and so a ^ 6 | a a1 and SU(a/a0 ) > SU(a/a1 ). 0 By the induction hypothesis applied to tp(a/a1 ) we obtain a2 , . . . , an and (αi : 0 < i < n) as required. a Definition 19.18. Let T be supersimple. For all hyperimaginaries a, e we define the analysability rank of a over e as the least ordinal α (if there is one) for which there is an analysis in n steps a0 , . . . , an of tp(a/e) as in Proposition 19.17 and the corresponding ordinals αi are ≤ α. We denote it by Ran (a/e). If there is no such ordinal, we set Ran (a/e) = ∞. By Proposition 19.17 Ran (a/e) < ∞ if SU(a/e) < ∞. Notice also that Ran (a/e) = 0 if and only if a ∈ bdd(e). Lemma 19.19. Let T be supersimple and let a, b, e be hyperimaginaries, and assume SU(a/e) < ∞.

148

19. Orthogonality and analysability

1. Ran (a/eb) ≤ Ran (a/e). 2. If a ^ | e b, then Ran (a/eb) = Ran (a/e). Proof. 1. We proceed by induction on SU(a/eb) and we may assume SU(a/e) > 0. We claim that for some α ≤ Ran (a/e) there is some d ∈ dclheq (aeb) r bdd(eb) such that tp(d/eb) is internal in Pα . Otherwise, whenever α ≤ Ran (a/e), d ∈ dclheq (aeb) and tp(d/eb) is internal in Pα , then d ∈ bdd(eb). Let a0 , . . . , an with ordinals (αi : i < n) be an analysis of tp(a/e) with αi ≤ Ran (a/e) for all i < n. Note that a0 = e ∈ bdd(eb). Assume ai ∈ bdd(eb). Since ai+1 ∈ dclheq (aeb) and tp(ai+1 /ai ) is internal in Pαi , by Lemma 19.10 tp(ai+1 /bdd(eb)) is internal in Pαi and by Lemma 19.8 tp(ai+1 /eb) is internal in Pαi . By induction, an ∈ bdd(eb). Since a ∈ bdd(an ), a ∈ bdd(eb) and therefore Ran (a/be) = 0. So, we may assume the claim is true. Choose â0 minimal such that there is some d ∈ dclheq (aeb) r bdd(eb) such that tp(d/eb) is internal in the family Pâ0 . By Corollary 19.15, tp(a/be) is orthogonal to every type of D-rank < â0 . Note that a ^ 6 | eb d and hence SU(a/ebd ) < SU(a/eb). By the induction hypothesis Ran (a/ebd ) ≤ Ran (a/e). A suitable analysis of tp(a/bde) after b0 = be and b1 = b0 d with first ordinal â0 gives an analysis of tp(a/be) with ordinals ≤ Ran (a/e). 2. The proof is by induction on SU(a/e). Let α = Ran (a/eb). We prove that Ran (a/e) ≤ α assuming a ^ | e b. Let a0 , . . . , an be an analysis of tp(a/eb) with ordinals (αi : i < n), all ≤ α. Assume for all â ≤ α, for all d ∈ bdd(ae) such that tp(d/e) is internal in Pâ , d ∈ bdd(e). We prove by induction 0 on i that ai ^ | e a. This is clear for a0 = be. Let ai+1 = Cb(ai+1 /ae). 0 Then ai+1 ∈ bdd(ae). Since tp(ai+1 /ai ) is internal in Pαi , by Lemma 19.11 0 tp(ai+1 /ai ) is internal in Pαi . By the induction hypothesis, ai ^ | e a. hence 0 0 ai+1 ^ | e ai and by Lemma 19.8 tp(ai+1 /e) is internal in Pαi . By our assump0 tion, ai+1 ∈ bdd(e) and therefore ai+1 ^ | e a. Since a ∈ bdd(an ), we conclude that a ^ | e a and hence a ∈ bdd(e) and Ran (a/e) = 0. On the other hand, if the assumption does not hold, then by Lemma 19.12 for some ordinal â ≤ α there is some d ∈ dcl(ae) r bdd(e) such that tp(d/e) is internal in Pâ . Choose â0 minimal with this property and choose a corresponding d . By Corollary 19.15 tp(a/e) is orthogonal to every type of D-rank < â0 . Note that a^ 6 | e d and therefore SU(a/ed ) < SU(a/e). Since a ^ | ed b, by the induction an an an hypothesis R (a/ed ) ≤ R (a/edb). By 1, R (a/edb) ≤ α. Adding to b0 = e and b1 = ed (and with first ordinal â0 ) an analysis of tp(a/ed ) with ordinals ≤ α we obtain an analysis of tp(a/e) with ordinals ≤ α. a Lemma 19.20. Let T be supersimple and let a, b, e be hyperimaginaries. Assume SU(a/e) < ∞. If b ∈ bdd(ae), then Ran (b/e) ≤ Ran (a/e). Proof. By induction on SU(b/e). Let α = Ran (a/e) and let a0 , . . . , an be an analysis of tp(a/e) with ordinals (αi : i < n) such that αi ≤ α for all

19. Orthogonality and analysability

149

i < n. Since a ∈ bdd(an ), if b ^ | e an then b ^ | e a. In this case b ∈ bdd(e) and therefore Ran (b/e) = 0. Consequently we may assume that there is a maximal i < n such that b ^ | e ai . We claim that there is some ordinal â ≤ α for which there is some d ∈ bdd(ai b) r bdd(ai ) such that tp(d/ai ) is internal in Pâ . Otherwise for all â ≤ α, if d ∈ bdd(ai b) and tp(d/ai ) is internal in Pâ , then d ∈ bdd(ai ). We know that tp(ai+1 /ai ) is internal in the family Pαi . 0 0 Let ai+1 = Cb(ai+1 /ai b). By Lemma 19.11 tp(ai+1 /ai ) is also internal in 0 0 Pαi . Since ai+1 ∈ bdd(ai b) and αi ≤ α, it is clear that ai+1 ∈ bdd(ai ) and hence ai+1 ^ | a b. Since ai ^ | e b, it follows that ai+1 ^ | e b, in contradiction i with the choice of ai . By Lemma 19.12 for some â ≤ α there is some d ∈ dclheq (ai b) r bdd(ai ) such that tp(d/ai ) is internal in Pâ . Let â0 ≤ α be minimal with respect to the existence of some d ∈ dclheq (ai b) r bdd(ai ) such that tp(d/ai ) is internal in Pâ0 . By Corollary 19.15 tp(b/ai ) is orthogonal to every type of D-rank < â0 . Since b ^ 6 | a d , SU(b/ai d ) < SU(b/ai ) and, i an by the induction hypothesis, R (b/ai d ) ≤ Ran (a/ai d ). By Lemma 19.19 Ran (a/ai d ) ≤ α. Adding an analysis of tp(b/ai d ) witnessing Ran (b/ai d ) ≤ α after the beginning b0 = ai and b1 = ai d (with first ordinal â0 ) we obtain an analysis of tp(b/ai ) with ordinals ≤ α showing Ran (b/ai ) ≤ α. Since b ^ | e ai , by Lemma 19.19 Ran (b/e) = Ran (b/ai ) ≤ α. a Proposition 19.21. Let T be supersimple and let a, e be hyperimaginaries. If SU(a/e) < ∞, then Ran (a/e) is the least ordinal α such that tp(a/e) is analysable in the family P≤α in finitely many steps. Proof. We assume that a0 , . . . , an ∈ dclheq (e, a), a0 = e, a ∈ bdd(an ), and for each i < n, ai ∈ dclheq (ai+1 ), tp(ai+1 /ai ) is internal in P≤α , and we prove that Ran (a/e) ≤ α. The proof is by induction on SU(a/e). We may clearly assume SU(a/e) > 0. We may also assume a1 6∈ bdd(e) since otherwise, by lemmas 19.8 and 19.10, tp(a2 /e) is internal in P≤α and hence we can delete a1 in the analysis. As in the proof of Proposition 19.17, tp(a/e) is nonorthogonal to some type of D-rank â for some ordinal â. Choose â minimal with this property. We claim that â ≤ α. Assume â > α. By Lemma 19.4, tp(a1 /e) is orthogonal to every type of D-rank < â. Since tp(a1 /e) is internal in P≤α , there is some set A such that e ∈ dclheq (A) and a1 ^ | e A and some tuple c of realizations of formulas of D-rank ≤ α over A such that a1 ∈ dclheq (A, c). We want to prove that a1 ^ | A c and hence we may assume c is a finite tuple, say c = c1 , . . . , cm . Since D(tp(cj /Ac<j )) ≤ α < â, by induction using orthogonality we get a1 ^ | A c. This implies a1 ^ | A a1 and therefore a1 ^ | e a1 and a1 ∈ bdd(e), contrarily to our assumption. Consequently, â ≤ α. By Corollary 19.15, there is some d ∈ dcl(ae) r bdd(e) such that tp(d/e) is internal in Pâ . It follows that a ^ 6 | e d and hence SU(a/de) < SU(a/e).

150

19. Orthogonality and analysability

Note that a0 d, a1 d, . . . , an d witnesses that tp(a/ed ) is analysable in P≤α in finitely many steps. By the induction hypothesis, Ran (a/ed ) ≤ α. Clearly, we can then start with b0 = e and b1 = ed (with first ordinal â0 = â) and continue with a suitable analysis of tp(a/ed ), say b1 , . . . , bk with ordinals (âi : 1 ≤ i < k) obtaining witnesses of Ran (a/e) ≤ α. a Lemma 19.22. Let T be supersimple and let a, b, e be hyperimaginaries. If SU(ab/e) < ∞, and Ran (a/e), Ran (b/ae) ≤ α, then Ran (ab/e) ≤ α. Proof. Let a0 , . . . , an be an analysis of tp(a/e) with ordinals ≤ α witnessing Ran (a/e) ≤ α, and let b0 , . . . , bm be a corresponding analysis witnessing Ran (b/Ae) ≤ α. Then ae ∈ bdd(an ), b0 = ae, and ab ∈ bdd(bm ). Take a set A such that b0 ∈ dclheq (A). Since an ∈ dclheq (A) and b0 ^ | a A, it n is clear that tp(b0 /an ) is internal in any family Pã . Hence the sequence a0 , . . . , an , b0 , . . . , bm shows that tp(ab/e) is analysable in P≤α . By Proposition 19.21, Ran (ab/e) ≤ α. a Lemma 19.23. Let T be supersimple and let a, b, e be hyperimaginaries. If SU(ab/e) < ∞, then Ran (ab/e) = max{Ran (a/e), Ran (b/e)}. Proof. By Lemma 19.20, Ran (a/e), Ran (b/e) ≤ Ran (ab/e). By lemmas 19.19 and 19.22, if Ran (a/e), Ran (b/e) ≤ α also Ran (ab/e) ≤ α. a Proposition 19.24. Let T be supersimple and let a, b, e be hyperimaginaries. If tp(b/e) is orthogonal to every type of D-rank < α and Ran (a/e) < α, then tp(b/e) ⊥ tp(a/e). Proof. Let d be a hyperimaginary such that e ∈ dclheq (d ) and b ^ | e d . We have to show that for every a 0 ≡e a such that a 0 ^ | e d we have b ^ | d a 0 . Note that, by Lemma 19.19, Ran (a 0 /d ) = Ran (a 0 /e) = Ran (a/e) < α. Hence, there is an analysis a0 , . . . , an of tp(a 0 /d ) with ordinals αi < α. In particular, a0 = d and a 0 ∈ bdd(an ). If b ^ | d an then b ^ | d a 0 and we have finished. Otherwise there is a maximal i such that b ^ | d ai . Then b ^ | e ai . Since tp(ai+1 /ai ) is internal in the family Pαi , there is a set A such that ai ∈ dclheq (A) and ai+1 ^ | a A and a tuple c of realizations of formulas over A of i

D-rank αi such that ai+1 ∈ dclheq (A, c). We may assume A ^ | a b. Then i+1 A^ | a bai+1 and thus b ^ | e A. We want to check that b ^ | A c, and to do i so we may assume c is a finite tuple, say c = c1 , . . . , cm , where for every j = 1, . . . , m, D(tp(cj /A)) < α. By our orthogonality assumption one easily sees by induction that b ^ | A c1 , . . . , cj for every j ≤ m. In particular, b ^ | A c. Consequently, b ^ | A ai+1 and b ^ | d ai+1 , a contradiction. a Proposition 19.25. Let T be supersimple, let e be a hyperimaginary, and let a = (ai : i ∈ I ) be a sequence of hyperimaginaries ai . If SU(a/e) < ∞ and Ran (ai /e) < α for each i ∈ I , then Ran (a/e) < α.

19. Orthogonality and analysability

151

Proof. Since SU(a/e) < ∞, by Proposition 16.27, there is a finite subset I0 ⊆ I such that a ∈ bdd(a  I0 , e). By Lemma 19.20 it is then enough to prove the result for I finite, and this can be easily done by induction on |I | using Lemma 19.23. a Proposition 19.26. Let T be supersimple, let a, e be hyperimaginaries and let a 0 be a hyperimaginary equivalent to some enumeration of D = {b ∈ dclheq (a, e) : Ran (b/e) < α}. If SU(a/e) < ∞, then Ran (a 0 /e) < α and tp(a/a 0 ) is orthogonal to every type of D-rank < α. Proof. Notice that SU(a 0 /e) < ∞. By Proposition 19.25, Ran (a 0 /e) < α. Assume now tp(a/a 0 ) is nonorthogonal to some type of D-rank â < α. By Corollary 19.15 there is some d ∈ dclheq (aa 0 ) r bdd(a 0 ) such that tp(d/a 0 ) is internal in Pâ . Let a0 , . . . , an be an analysis of tp(a 0 /e) with ordinals < α. Since a 0 ∈ bdd(an ), by lemmas 19.8 and 19.10 tp(an d/an ) is internal in Pâ . The tuples an and an d provide an analysis of tp(d/an ) in Pâ . By Proposition 19.21, Ran (d/an ) < α. Since also Ran (an /e) < α, by Lemma 19.22, Ran (dan /e) < α. Since a 0 ∈ dcl(ae), we have dan ∈ dcl(ae). Hence dan ∈ D and dan ∈ dcl(a 0 ), in contradiction with d 6∈ bdd(a 0 ). a

Chapter 20

HYPERIMAGINARIES IN SUPERSIMPLE THEORIES

Lemma 20.1. Let T be supersimple, let E = E(x, y) be a 0-type-definable equivalence relation and let q(x) ∈ S(∅). There is a formula ø(x, y) ∈ E(x, y) such that for all a, b |= q, if |= ø(a, b) then bdd(aE ) = bdd(bE ). Proof. As usual, we assume that all formulas in E(x, y) are symmetric. Let ϕ(x, y) ∈ E(x, y) be a formula such that ϕ(x, a) has smallest D-rank among all formulas in E(x, y) for a |= q. Choose ø(x, y) ∈ E(x, y) such that ø 3 (x, y) ` ϕ(x, y), that is, ø(x, z) ∧ ø(z, u) ∧ ø(u, y) ` ϕ(x, y). Assume a, b |= q, |= ø(a, b) and bE 6∈ bdd(aE ). By Proposition 16.4, there is an aE -indiscernible sequence ([bi ]E : i < ù) of distinct hyperimaginaries [bi ]E starting with [b0 ]E = bE . We can extend the sequence and apply the ˝ Erdos–Rado Theorem to obtain a formula ÷(x, y) ∈ E such that |= ¬÷(bi , bj ) for all i < j < ù. We can assume b = b0 and b ≡aE bi for all i < ù. Choose è(x, y) ∈ E(x, y) such that è(x, y) ` ø(x, y) and è 2 (x, y) ` ÷(x, y). For each i < ù there is some ai such that E(ai , a) and ab ≡ ai bi . Hence |= ø(bi , ai ) and therefore è(x, bi ) ` ϕ(x, a). Moreover è(x, bi ) ∧ è(x, bj ) is inconsistent for all i < j < ù. We can extend the sequence (bi : i < ù) and then apply Proposition 1.6, so we can assume (bi : i < ù) is indiscernible over a. Then è(x, b0 ) divides over a, and by Remark 14.2, D(è(x, b0 )) < D(ϕ(x, a)). Since b0 |= q, this contradicts the minimality in the choice of ϕ(x, y). a Lemma 20.2. Assume T is supersimple, and α is an ordinal number such that for every hyperimaginary a0 , if there exists a sequence of imaginaries a with SU(a) < ∞ and such that a0 ∈ dclheq (a), and Ran (a/a0 ) < α, then a0 is eliminable. If a0 is a hyperimaginary and there is a sequence of imaginaries a such that SU(a) < ∞, and a0 ∈ dclheq (a), and tp(a/a0 ) is internal in the family Pα of all formulas of D-rank α and it is orthogonal to every type of D-rank < α, then a0 is eliminable. 153

154

20. Hyperimaginaries in supersimple theories

Proof. We will work in T eq , so acl = acleq . We start by changing a0 by a more convenient hyperimaginary aE . By propositions 15.6, and 17.8, there is a 0-type-definable equivalence relation E such that tp(a/aE ) is an amalgamation base, aE ∈ bdd(a0 ), and a0 ∈ dclheq (aE ). Note that a ^ | a aE . By Proposition 19.3 tp(a/aE ) is or0 thogonal to every type of D-rank < α and by Lemma 19.8 tp(a/aE ) is internal in Pα . By Lemma 18.6 it suffices to show that aE is eliminable. ˜ < ∞, aE ∈ We will find a sequence of imaginaries a˜ such that SU(a) ˜ and Ran (a/a ˜ E ) < α. The assumption of the lemma will imply then dclheq (a), that aE is eliminable and the proof will be completed. Let q(x) = tp(a). By Remark 15.17 tp(a/aE ) ≡ q(x) ∧ E(x, a). Since SU(a) < ∞, by Proposition 16.27 there is a finite subtuple a − of a such that a ⊆ acl(a − ). If a = (ai : i ∈ I ), there is a finite subset I − ⊆ I such that a − = (ai : i ∈ I − ). For any other sequence b = (bi : i ∈ I ) we will understand that b − = (bi : i ∈ I − ). In particular, if x is a tuple of variables corresponding to a, then x − is the finite subtuple of x that corresponds to a − . We can choose a partial type Σ over ∅ which defines E  q, the restriction of E to realizations of q, and has the following additional properties: 1. If ø(x, y) ∈ Σ, then ø(x, y) ` xi ∈ acl(x − ) and ø(x, y) ` yi ∈ acl(y − ) for all variables xi , yi appearing in ø. 2. All formulas ø(x, y) ∈ Σ are symmetric. 3. If ø0 (x, y) ∈ E is as in Lemma 20.1 for E and q, then ø(x, y)∧ø(y, z) ` ø0 (x, z) for all ø(x, y) ∈ Σ(x, y). By Lemma 19.10 tp(a − /aE ) is internal in Pα and by Lemma 19.4 it is orthogonal to all types of D-rank < α. By internality, there is some set A such that aE ∈ dclheq (A) and for some n < ù there are formulas ϕ1 (z1 ), . . . , ϕn (zn ) over A of D-rank α and realizations c1 |= ϕ1 , . . . , cn |= ϕn such that a − ^ | a A E − − and a ∈ dcl(A, c1 , . . . , cn ). There is some formula ÷ = ÷(x , z1 , . . . , zn ) ∈ L(A) such that |= ∀z1 . . . zn ∃=1 x − ÷(x − , z1 , . . . , zn ) and |= ÷(a − , c1 , . . . , cn ). We define ÷0 (x − ) = ∃z1 . . . zn (÷(x − , z1 . . . , zn ) ∧ ϕ1 (z1 ) ∧ · · · ∧ ϕn (zn )). Let a¯ be the tuple a extended by finitely many parameters from A such ¯ Again, for each tuple that each ϕi (x − ) is over a¯ and ÷ is also over a. b = (bi : i ∈ I ) of the length of a, by b¯ we understand a corresponding tuple ¯ a finite extension of b. Let q( ¯ and let us write of the length of a, ¯ x) ¯ = tp(a) ¯ and ϕi (zi ) = ϕi (zi , a) ¯ with ÷0 (x − , y), ÷0 (x − ) = ÷0 (x − , a), ¯ ϕi (zi , y) ¯ ∈ L. We define D as the following class of hyperimaginaries: ¯ : Ran (d/aE ) < α}. D = {d ∈ dclheq (a) There is some hyperimaginary interdefinable with D and by Proposition 15.6 there is some 0-type-definable equivalence relation F such that a¯F is equivalent

20. Hyperimaginaries in supersimple theories

155

to some sequence enumerating D. By Proposition 19.26 Ran (a¯F /aE ) < α. By Lemma 19.20, Ran (d/aE ) < α if d ∈ bdd(a¯F ). It follows that ¯ ∩ bdd(a¯F ). D = dclheq (a) ¯ a¯F ) is an amalgamation base. Since a¯ is a finite By Proposition 17.8, tp(a/ extension of a, it is contained in the algebraic closure of a finite tuple and ¯ < ∞. Since aE ∈ D (we may by Remark 16.26 and supersimplicity, SU(a) heq ¯ a¯F ) assume α > 0) it is clear that aE ∈ dcl (a¯F ). By Proposition 19.26 tp(a/ is orthogonal to every type of D-rank < α. We recapitulate: ¯ a¯F ) is an amalgamation base. 1. tp(a/ ¯ < ∞. 2. SU(a) 3. aE ∈ dclheq (a¯F ). 4. Ran (a¯F /aE ) < α. ¯ a¯F ) is orthogonal to every type of D-rank < α. 5. tp(a/ All these properties still hold for any realization a¯ |= q¯ and therefore we can forget about the particular realization of q¯ we were discussing. We define a relation Sø = Sø (x, ¯ y) for every ø(x, y) ∈ Σ as follows: ¯ b) if and only if a¯ |= q, Sø (a, ¯ and b |= q, and there is some c¯ |= q¯ such that ¯ a) ¯ and c¯ ^ • F (c, | a¯ b, F ¯ ∧ ø(x, c) ∧ ø(x, b) does not fork over a¯F . • |= ø0 (c, b) and ÷0 (x − , c) We want to prove that Sø is definable on q¯ × q, that is, there is a formula ¯ b) if and only if ñø (x, ¯ y) ∈ L such that for a¯ |= q, ¯ for b |= q, |= ñø (a, ¯ b). For this it is enough to prove that Sø and its complement Sø are Sø (a, type-definable on q¯ × q. Claim 1. Sø is type-definable on q¯ × q. ¯ b) if and Proof of Claim 1. Let a¯ |= q¯ and b |= q. By definition, Sø (a, ¯ a), ¯ c¯ ^ ¯ ∧ ø(x, c) ∧ only if for some c¯ |= q, ¯ F (c, | a¯ b, |= ø0 (c, b) and ÷0 (x − , c) F ø(x, b) does not fork over a¯F . Since a¯F = c¯F and (by Remark 15.17) ¯ a¯F ) ≡ q( ¯ is constant for every a¯ |= q, tp(a/ ¯ x) ¯ ∧ F (x, ¯ a) ¯ the condition c¯ ^ | a¯ b F is type-definable by Corollary 16.41. We must check that ¯ ∧ ø(x, c) ∧ ø(x, b) does not fork over a¯F ÷0 (x − , c) ¯ b (recall a¯F = c¯F ). By Proposiis also type-definable as a condition on c, tion 16.23 it is equivalent to the existence of a Morley sequence (c¯i , bi : i < ù) ¯ b/c¯F ) such that {÷0 (x − , c¯i ) ∧ ø(x, ci ) ∧ ø(x, bi ) : i < ù} is consisin tp(c, ¯ b/c¯F ) is a c¯F -indiscernible, c¯F -independent tent. A Morley sequence in tp(c, ¯ b/c¯F ). Since tp(c, ¯ b/c¯F ) is not fixed for all sequence of realizations of tp(c, c¯ |= q¯ and b |= q, we cannot use Corollary 16.42. Indiscernibility is not a problem. Notice that (c¯i bi : i < ù) is independent over c¯F if and only if c¯i ^ | c¯ c¯
156

20. Hyperimaginaries in supersimple theories

assuming a¯F = c¯F and |= ø0 (a, b) we can express bi ^ | b
by a partial type. Notice that, on q, ¯ F implies E and then aE = cE . Since |= ø0 (a, b), the equivalence classes aE and bE are interbounded. By Lemma 19.19 Ran (a¯F /bE ) < α and therefore, by Proposition 19.24, tp(b/bE ) ⊥ tp(a¯F /bE ). By Proposition 19.5, tp((bi : i < ù)/bE ) ⊥ tp(a¯F /bE ). Hence bi b
E

Claim 2. Sø is type-definable on q¯ × q. ¯ b) if Proof of Claim 2. Let a¯ |= q¯ and b |= q. Notice first that Sø (a, ¯ c), ¯ c¯ ^ and only if there exists some c¯ |= q¯ such that F (a, | a¯ b and either F ¯ ∧ ø(x, c) ∧ ø(x, b) forks over a¯F . We check this. 6|= ø0 (c, b) or ÷0 (x − , c) ¯ b) and that there is some c¯ as described. By definition Assume first Sø (a, ¯ d¯ ^ of Sø there is another d¯ |= q¯ such that F (d¯, a), | a¯ b, |= ø0 (d, b), and F Ls ÷0 (x − , d¯) ∧ ø(x, d ) ∧ ø(x, b) does not fork over a¯F . Since c¯ ≡a¯F d¯, by ¯ ∧ ø(x, c) ∧ ø(x, b) does not fork over a¯F and hence Corollary 16.38 ÷0 (x − , c) 6|= ø0 (c, b). But ø(x, c) ∧ ø(x, b) is consistent and by assumption ø(x, y) is symmetric and ø(x, y) ∧ ø(y, z) ` ø0 (x, z), which implies |= ø0 (c, b), a contradiction. The other direction is clear. Condition c¯ ^ | a¯ b (assuming a¯F = c¯F ) is easily seen to be type-definable F by Corollary 16.41. A more difficult task is to show that condition ¯ ∧ ø(x, c) ∧ ø(x, b) forks over a¯F ÷0 (x − , c) is type-definable. We can solve the problem proving that in fact, assuming ¯ c), ¯ and c¯ ^ ¯ ∧ ø(x, c) ∧ ø(x, b) |= ø0 (b, c), F (a, | a¯ b, the formula ÷0 (x − , c) F forks over a¯F if and only if it n + 2-divides over a¯F (recall that n is the number of formulas witnessing internality). Clearly n + 2-dividing over a¯F is type-definable over any representative of a¯F . ¯ c), ¯ c¯ ^ ¯ ∧ ø(x, c) ∧ ø(x, b) We assume |= ø0 (b, c), F (a, | a¯ b, and ÷0 (x − , c) F forks over a¯F and we will prove that this formula n + 2-divides over a¯F . This will finish the proof of the claim. Let (c¯i bi : i < ù) be a Morley sequence in ¯ a¯F ). It will suffice to check inconsistency of tp(cb/ {÷0 (x − , c¯i ) ∧ ø(x, ci ) ∧ ø(x, bi ) : i < n + 2}. Suppose it is consistent, and let e realize it. Let 0 < i < n + 2. Then e^ 6 | a¯ c¯i bi . Since c¯i bi ^ | a¯ c¯0 and a¯F ∈ dclheq (c¯i ), we get e ^ 6 | c¯ c¯i bi . The F F 0 formulas ϕ1 (z1 , c¯1 ), . . . , ϕn (zn , c¯n ) have D-rank α and, since |= ÷(e − , c¯0 ),

20. Hyperimaginaries in supersimple theories

157

there are dj |= ϕj (zj , c¯0 ) such that e − ∈ dcl(c¯0 , d1 , . . . , dn ). Since ø(x, y) ∈ Σ and |= ø(e, ci ), we know that e ⊆ acl(e − ). Hence d1 , . . . , dn ^ 6 | c¯ c¯i bi . 0 Now we claim that tp(c¯i bi /c¯0 ) is orthogonal to all types of D-rank < α if i > 0. Note that c¯i ^ | c¯ bi (because a¯F ∈ dclheq (c¯0 ), and c¯i ^ | a¯ bi , and F 0 c¯i bi ^ | a¯ c¯0 ) and then, by Proposition 19.3, it is enough to prove that tp(c¯i /c¯0 ) F and tp(bi /c¯0 ) are orthogonal to all types of D-rank < α. Orthogonality of tp(c¯i /c¯0 ) is clear by Proposition 19.3 since a¯F ∈ dclheq (c¯0 ), and c¯i ^ | a¯ c¯0 , F and tp(c¯i /a¯F ) is orthogonal to all types of D-rank < α. With respect to tp(bi /c¯0 ), notice that since tp(bi /bE ) is orthogonal to all types of D-rank < α and Ran (a¯F /bE ) < α, by Proposition 19.22 tp(bi /bE ) ⊥ tp(a¯F /bE ) and therefore bi ^ | b a¯F . Then tp(bi /c¯0 ) is also orthogonal to all types of D-rank E < α, because bi ^ | a¯ c¯0 and thus bi ^ | b c¯0 . F E Lemma 19.6 implies that d1 , . . . , dn ^ 6 | c¯ c¯i bi for at most n tuples c¯i bi , a 0 contradiction. a By the claims, there is a formula ñø (x, ¯ y) ∈ L such that for all a¯ |= q, ¯ b |= q, ¯ b) if and only if Sø (a, ¯ b). |= ñø (a, ¯ c¯ |= q¯ such that F (a, ¯ c), ¯ Note that for all b |= q and all a, ¯ b) if and only if Sø (c, ¯ b). Sø (a, Therefore there is a formula ϕø (z) ∈ q(z) such that q( ¯ x) ¯ ∧ q( ¯ y) ¯ ∧ F (x, ¯ y) ¯ ` ∀z(ϕø (z) → (ñø (x, ¯ z) ↔ ñø (y, ¯ z))). The formula ∀z(ϕø (z) → (ñø (x, ¯ z) ↔ ñø (y, ¯ z))) defines an equivalence relation Eø . Note that a¯Eø ∈ dclheq (a¯F ) and it is in fact an imaginary. V Claim 3. q( ¯ x) ¯ ∧ q( ¯ y) ¯ ∧ ø∈Σ Eø (x, ¯ y) ¯ ` E(x, y). ¯ ¯ ¯ b) Proof of Claim 3. Assume ¯ We check that Eø (a, V a¯ |= q¯ and b |= q. ¯ b) and that ø∈Σ Sø (a, ¯ b) implies E(a, b). The second stateimplies Sø (a, ment is clear since on realizations of q, ¯ F (x, ¯ y) ¯ ` E(x, y) and since for each ø(x, y) ∈ Σ we may find some ø 0 (x, y) ∈ Σ such that ø 0 (x, z) ∧ ø 0 (z, u) ∧ ø 0 (u, y) ` ø(x, y). ¯ a) whenever a¯ |= q. For the first statement it is enough to prove that Sø (a, ¯ ¯ a) ¯ and c¯ ^ ¯ Then E(c, a) and To do so, choose c¯ |= q¯ such that F (c, | a¯ a. F therefore |= ø0 (c, a). It suffices then to check that ¯ ∧ ø(x, c) ∧ ø(x, a) does not fork over a¯F . ÷0 (x − , c) Let c + be the finite subtuple of c¯ consisting of all its elements that do not occur in c. Then c¯ = cc + and c ^ | c c + . Choose d ≡c + c such that E(d, a) and E

158

20. Hyperimaginaries in supersimple theories

¯ Then, by conjugation, d ^ ¯ a¯ c. | a c + and by transitivity d ^ | a a¯ c. E E heq heq ¯ we get d ^ ¯ Moreover Since aE ∈ dcl (a¯F ) and a¯F ∈ dcl (a), | a¯ a¯ c. F − − + ¯ ∧ ø(d, c) ∧ ø(d, a) because ÷0 (x , c) ¯ is over c , |= ÷0 (c − , c), ¯ |= ÷0 (d , c) and dE = aE = cE . a d^ | a

Ec

+

Now consider the sequence of imaginaries a˜ = (a¯Eø : ø ∈ Σ). ˜ and a˜ ∈ dclheq (a¯F ). By Lemma 19.20 Ran (a/a ˜ E) ≤ Note that aE ∈ dclheq (a) ˜ < ∞ since Ran (a¯F /aE ) and we know that Ran (a¯F /aE ) < α. Finally SU(a) ¯ < ∞. Thus a˜ satisfies all the required conditions and the proof of the SU(a) lemma finishes. a Remark 20.3. A simplified version of Lemma 20.2 can be used to offer a different proof of Theorem 18.29 on elimination of bounded hyperimaginaries in low theories. Since the hyperimaginaries are bounded, no application of Lemma 20.1 is needed. Moreover the extension of a to a¯ is not necessary since we can use type-definability of forking to justify easily the type-definability of Sø . Hence it is enough to define Sø on realizations of q(x) by: Sø (a, b) if and only if there is some c |= q such that • E(c, a) and c ^ | a b, E • ø(x, c) ∧ ø(x, b) does not fork over aE . The hyperimaginary aE turns out to be equivalent to the sequence of imaginar˜ ies a. Theorem 20.4. Supersimple theories eliminate hyperimaginaries. Proof. By Corollary 18.11 it suffices to show that T eliminates finitary hyperimaginaries. In fact we will prove a slightly stronger form of elimination: whenever a hyperimaginary a0 is definable over an imaginary tuple a such that SU(a) < ∞, then a0 is eliminable. The proof is by induction in Ran (a/a0 ). Assume that Ran (a/a0 ) = α and the theorem holds for all a 0 , a00 with Ran (a 0 /a00 ) < α. Since Ran (a/a0 ) = α, for some n < ù there are hyperimaginaries a1 , . . . , an ∈ dclheq (a) and a sequence of ordinal numbers (αi : i < n) such that a ∈ bdd(an ), ai ∈ dclheq (ai+1 ), αi ≤ α, tp(ai+1 /ai ) is internal in the family of all formulas of D-rank αi , and tp(a/ai ) is orthogonal to every type of D-rank < αi . By Lemma 18.6, an is equivalent to a sequence of imaginaries and we can replace it by this sequence and assume an is a sequence of imaginaries. Moreover SU(an ) < ∞ since SU(a) < ∞ and an is definable over a. Note also that a0 ∈ dclheq (an ) and Ran (an /a0 ) ≤ α. We may proceed by a second induction on n. Note that Ran (an /an−1 ) ≤ αn−1 ≤ α. If αn−1 < α we can use the general induction hypothesis to eliminate an−1 . If αn−1 = α we can use Lemma 20.2 to eliminate

20. Hyperimaginaries in supersimple theories

159

an−1 again. By the induction hypothesis on the length of the analysis, we finish. a Corollary 20.5. Let T be supersimple. Then Ls

s

1. a ≡A b if and only if a ≡A b. 2. If A = acleq (A), then every p(x) ∈ S(A) is an amalgamation base. Proof. 1 is clear since by elimination of hyperimaginaries we have Aut(C/bdd(A)) = Aut(C/acleq (A)). For 2 notice that for any a, tp(a/A) ` tp(a/bdd(A)). a Definition 20.6. Let p(x) be an amalgamation base. We say that a formula ϕ(x, y) ∈ L is p-stable if p  ϕ = q  ϕ for all p, q ∈ Pp . Lemma 20.7. Let T be simple and let p(x) be an amalgamation base. If ϕ(x, y) ∈ L is p-stable then for every p ∈ Pp , p  ϕ is definable. Proof. We may assume p(x) ∈ S(A) for a set A of real elements. By Corollary 5.23 there is a set of formulas over A, ð(x), such that for all a, a |= ð if and only if p(x) ∪ {ϕ(x, a)} does not fork over A, and similarly there is a set of formulas over A, ð0 (x), such that all a, a |= ð0 if and only if p(x) ∪ {¬ϕ(x, a)} does not fork over A. By the uniqueness of the ϕ-type of any p ∈ Pp , ð(x) ∪ ð0 (x) is inconsistent. Choose ø(x) ∈ ð(x) such that ø(x) is inconsistent with ð0 (x). Then ø(x) is the definition we wanted. a Proposition 20.8. Let T be simple. A formula ϕ(x, y) ∈ L is stable if and only if it is p-stable for every amalgamation base p(x). Proof. Assume ϕ(x, y) is not p-stable for some amalgamation base p(x) ∈ S(A). Then for some tuple a both p(x) ∪ {ϕ(x, a)} and p(x) ∪ {¬ϕ(x, a)} do not fork over A. Let (ai : i < ù) be a Morley sequence in tp(a/A). Since p(x) is an amalgamation base and the sequence is A-independent, for every X ⊆ ù, p(x) ∪ {ϕ(x, ai ) : i ∈ X } ∪ {¬ϕ(x, ai ) : i ∈ ù r X } is consistent. Then |Sϕ ({ai : i < ù})| ≥ 2ù and therefore ϕ(x, y) is unstable. Assume now that ϕ(x, y) is unstable. There is an indiscernible sequence (ai bi : i < ù + ù) such that |= ϕ(ai , bj ) if and only if i < j. Let A = {ai bi : i < ù}. By Corollary 17.7 p(x) = tp(aù /A) is an amalgamation base and by Lemma 10.4 (ai bi : ù ≤ i < ù + ù) is a Morley sequence over A. Since aù+2 |= ¬ϕ(x, bù+1 ) and aù+2 ^ | A bù+1 , p(x) ∪ {¬ϕ(x, bù+1 )} does not fork over A. Since aù |= ϕ(x, bù+1 ) and aù ^ | A bù+1 , p(x) ∪ {ϕ(x, bù+1 )} does not fork over A. This means that ϕ(x, y) is not p-stable. a Definition 20.9. Let p(x) be an amalgamation base and let ϕ(x, y) ∈ L. We say that ϕ(x, a) is a canonical formula for p if ϕ(x, a) ∈ p for some p ∈ Pp and for every a 0 ≡ a, if ϕ(x, a 0 ) ∈ p for some p ∈ Pp then ϕ(x, a) ≡ ϕ(x, a 0 ).

160

20. Hyperimaginaries in supersimple theories

Theorem 20.10. Let T be supersimple. For every amalgamation base p(x), Cb(p) is equivalent to the set of canonical parameters of all canonical formulas for p. Proof. By Theorem 20.4 we know that T eliminates hyperimaginaries and therefore Cb(p) is equivalent to a sequence of imaginaries c. Since p  c is an amalgamation base, we may assume p(x) ∈ S(c). It is clear that all canonical parameters of canonical formulas for p are definable over c. By supersimplicity there is a finite subtuple c0 of c such that p does not fork over c0 . By Lemma 17.23, c ∈ bdd(c0 ) and since they are tuples of imaginaries, c ⊆ acleq (c0 ). We show now that every finite subtuple c1 of c extending c0 is a canonical parameter of some canonical formula for p. For any tuple d of the length of c we let d1 be the corresponding finite subtuple of d . We write p(x) = p(x, c) with p(x, z) ∈ S(∅). Notice that for any c 0 ≡ c, c = c 0 if and only if p(x, c) ∪ p(x, c 0 ) does not fork over c. Hence, if c 0 ≡ c and c10 6= c1 , then p(x, c) ∪ p(x, c 0 ) forks over c. By Proposition 5.22 and Remark 3.14 there are formulas ϕ(x, y), ø(x, z2 ) ∈ L, a finite subtuple c2 of c extending c1 , and a natural number k such that for all c20 ≡ c2 , c1 6= c10 implies D(ø(x, c2 ) ∧ ø(x, c20 ), ϕ, k) < D(p(x, c), ϕ, k). Let n = D(p(x, c), ϕ, k). Extending ø(x, z2 ) and c2 if necessary, we may assume that D(ø(x, c2 ), ϕ, k) = n. Since c2 is algebraic over c1 , there is a formula ÷(z2 , z) ∈ L such that |= ÷(c2 , c1 ) and ÷(z2 , c1 ) ` tp(c2 /c1 ). Let ø 0 (x, z) = ∃z2 (ø(x, z2 ) ∧ ÷(z2 , z)). Since c2 is algebraic over c1 , ø 0 (x, c1 ) is equivalent to a finite disjunction of c1 conjugates of ø(x, c2 ) and then, by Proposition 3.12, D(ø 0 (x, c1 ), ϕ, k) = n. For the same reason, for all c10 ≡ c1 , c10 6= c1 implies D(ø 0 (x, c10 ) ∧ ø 0 (x, c1 ), ϕ, k) < n. It follows that ø 0 (x, c1 ) is a canonical formula for p(x) and that c1 is its canonical parameter. a Proposition 20.11. Let T be simple and let p(x) be an amalgamation base. If Cb(p) is equivalent to the set of canonical parameters of all canonical formulas for p, then Cb(p) is also equivalent to the set of canonical parameters of the definitions of all p  ϕ for all p ∈ Pp for all p-stable formulas ϕ(x, y). Proof. The canonical base of p is equivalent to a tuple of imaginaries c. We may assume p(x) ∈ S(c). For any p-stable ϕ(x, y) ∈ L we know, by Lemma 20.7, that the common ϕ-type of any p ∈ Pp is definable. Let cϕ be the canonical parameter of such a definition. It is clear that each cϕ is definable over c. Now we show that each canonical parameter d of each

20. Hyperimaginaries in supersimple theories

161

canonical formula for p is in fact the canonical parameter cϕ of some p-stable formula ϕ(x, y). Let ø(x, y) ∈ L and assume that ø(x, d ) is canonical for p with canonical parameter d . Let r(y) = tp(d ). If d 0 |= r(y) and d 6= d 0 , then ø(x, d ) 6≡ ø(x, d 0 ) and therefore p(x) ∪ {ø(x, d 0 )} forks over c. By Corollary 5.23, there is some formula è(y) ∈ r(y) such that for all d 0 |= è(y), if d 6= d 0 , then p(x) ∪ {ø(x, d 0 )} forks over c. Let ϕ(x, y) = ø(x, y) ∧ è(y). Then ϕ(x, d 0 ) ∈ p for some p ∈ Pp if and only if d = d 0 . The only positive instance of ϕ(x, y) in a type in the amalgamation class Pp is ϕ(x, d ). Note that ¬ϕ(x, d ) does not appear in any such type since it is a formula over c and all types in Pp are extensions of p. This implies that ϕ(x, y) is p-stable and d ∼ cϕ . a Corollary 20.12. Let T be supersimple. For every amalgamation base p(x), Cb(p) is equivalent to the sequence of all canonical parameters of the definitions of all p  ϕ for all p ∈ Pp for all p-stable formulas ϕ(x, y). Proof. By Theorem 20.10 and Proposition 20.11. a

REFERENCES

[1] Hans Adler, Strict orders prohibit elimination of hyperimaginaries, Preprint, February 2007. [2] , A geometric introduction to forking and thorn-forking, Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 1–20. [3] , Thorn-forking as local forking, Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 21–38. [4] John T. Baldwin, Fundamentals of Stability Theory, Springer-Verlag, Berlin, 1988. [5] Itay Ben-Yaacov, Anand Pillay, and Evguenii Vassiliev, Lovely pairs of models, Annals of Pure and Applied Logic, vol. 122 (2003), pp. 235–261. [6] Steven Buechler, Lascar strong types in some simple theories, The Journal of Symbolic Logic, vol. 64 (1999), pp. 817–824. [7] Steven Buechler, Anand Pillay, and Frank O. Wagner, Supersimple theories, Journal of the American Mathematical Society, vol. 14 (2000), pp. 109–124. [8] Enrique Casanovas, Simplicity simplified, Revista Colombiana de Matem´aticas, vol. 41 (2007), pp. 263–277. [9] Enrique Casanovas and Byunghan Kim, A supersimple nonlow theory, Notre Dame Journal of Formal Logic, vol. 39 (1998), pp. 507–518. [10] Enrique Casanovas, Daniel Lascar, Anand Pillay, and Martin Ziegler, Galois groups of first order theories, Journal of Mathematical Logic, vol. 1 (2001), pp. 305–319. [11] Enrique Casanovas and Frank O. Wagner, The free roots of the complete graph, Proceedings of the American Mathematical Society, vol. 132 (2004), no. 5, pp. 1543–1548. [12] Bradd Hart, Byunghan Kim, and Anand Pillay, Coordinatisation and canonical bases in simple theories, The Journal of Symbolic Logic, vol. 65 (2000), pp. 293–309. [13] Wilfrid Hodges, A Shorter Model Theory, Cambridge University Press, Cambridge, 1997. [14] Ehud Hrushovski, Kueker’s conjecture for stable theories, The Journal of Symbolic Logic, vol. 54 (1989), pp. 207–220. 163

164

References

[15] Byunghan Kim, Simple First Order Theories, Ph.D. thesis, University of Notre Dame, June 1996. [16] , Forking in simple unstable theories, Journal of the London Mathematical Society, vol. 57 (1998), pp. 257–267. [17] , A note on Lascar strong types in simple theories, The Journal of Symbolic Logic, vol. 63 (1998), pp. 926–936. [18] , Simplicity, and stability in there, The Journal of Symbolic Logic, vol. 66 (2001), pp. 822–836. [19] Byunghan Kim and Anand Pillay, Simple theories, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 149–164. [20] , From stability to simplicity, The Bulletin of Symbolic Logic, vol. 4 (1998), pp. 17–36. [21] , Around stable forking, Fundamenta Mathematicae, vol. 170 (2001), pp. 107–118. [22] Daniel Lascar, On the category of models of a complete theory, The Journal of Symbolic Logic, vol. 47 (1982), pp. 249–266. [23] , Stability in Model Theory, Longman Scientific & Technical, Harlow, U.K., 1987. [24] , La th´eorie des mod`eles en peu de maux, Nouvelle biblioth`eque math´ematique, no. 10, Cassini, Paris, 2009. [25] Daniel Lascar and Anand Pillay, Forking and fundamental order in simple theories, The Journal of Symbolic Logic, vol. 64 (1999), pp. 1155–1158. [26] , Hyperimaginaries and automorphism groups, The Journal of Symbolic Logic, vol. 66 (2001), pp. 127–143. [27] Mihaly ´ Makkai, A survey of basic stability theory with particular emphasis on orthogonality and regular types, Israel Journal of Mathematics, vol. 49 (1984), pp. 181–238. [28] David Marker, Model Theory: An Introduction, Graduate Texts in Mathematics, vol. 217, Springer, New York, 2002. [29] Ludomir Newelski, The diameter of a Lascar strong type, Fundamenta Mathematicae, vol. 176 (2003), no. 2, pp. 157–170. [30] Ludomir Newelski and Marcin Petrykowski, Weak generic types and covering of groups I, Fundamenta Mathematicae, vol. 191 (2006), pp. 201– 225. [31] Rodrigo Pelaez, About the Lascar Group, Ph.D. thesis, University of ´ Barcelona, April 2008. [32] Anand Pillay, Forking, normalization and canonical bases, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 61–81. [33] , Geometric Stability Theory, Oxford University Press, 1996. [34] , Definability and definable groups in simple theories, The Journal of Symbolic Logic, vol. 63 (1998), pp. 788–796. [35] Anand Pillay and Bruno Poizat, Pas d’imaginaires dans l’infini!, The Journal of Symbolic Logic, vol. 52 (1987), no. 2, pp. 400– 403.

References

165

[36] Bruno Poizat, Cours de th´eorie des mod`eles, Nur al-Mantiq walMa’rifah, 82, rue Racine 69100 Villeurbanne, France, 1985, Diffus´e par OFFILIB. [37] Ziv Shami, Definability in low simple theories, The Journal of Symbolic Logic, vol. 65 (2000), pp. 1481–1490. [38] Saharon Shelah, Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177–203. [39] , Classification Theory, second ed., North Holland P.C., Amsterdam, 1990. [40] Katrin Tent and Martin Ziegler, A Course in Model Theory, Lecture Notes in Logic, Cambridge University Press, Cambridge, to appear. [41] Frank O. Wagner, Simple Theories, Mathematics and Its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000. [42] Martin Ziegler, Stabilit¨atstheorie, Freiburger Vorlesung gehalten im ¨ Wintersemester 1988/1989. Ausgearbeitet von Urs Kunzi, Oktober 1991.

INDEX

N

i∈I pi , 73 | -Morley sequence, 80 ^ | -free extension, 80 ^ | -independent sequence, 80 ^ | -stationary type, 82 ^ | ∗ , 77 ^

| d , 78 ^ | f , 75, 78 ^ 0-definable, 4 0-type-definable, 4 KP

a ≡ A b, 59 Ls

a ≡e b, 116 Ls

a ≡A b, 53 s a ≡A b, 59 a ∼ b, 101 acl(A), 3 aclheq (A), 102 acleq (A), 6 A-bounded hyperimaginary, 103 A-conjugate, 3 A-definable, 4 A-hyperimaginary, 101 A-imaginary, 5 A-independent sequence, 32 A-invariant, 3 A^ | C B, 31 A^ 6 | C B, 31 Autf(C/A), 55 Autf(C/e), 116 bdd(A), 102 CB∆ (ð), 37 Cb(a/A), 72 Cb(a/b), 128 Cb(p), 72, 128 Ceq , 5 Cheq , 101

dcl(A), 3 dclheq (A), 101 dcleq (A) , 6 diamA (X ), 55 dA (a, b), 55 dp xϕ(x, y), 11 DM, 96 D(ð, ∆, k), 19 D(ð, ϕ), 137 D(ð, ϕ, k), 19 D(ϕ), 93 D-rank, 93 ∆-formula, 19 ∆-multiplicity, 37 ∆-type, 19 e-hyperimaginary, 117 e ≡c d , 103 k-dividing, 21 k-inconsistent, 17 k-tree property, 17 κ(T ), 85 Lstp(a/A), 53 Lstp(a/e), 116 ë-stable formula, 12 theory, 17 M -special, 43 Mlt(p), 69 Mlt∆ (ð), 37 (M, dp), 45 ncnA (x, y), 54 ncA (x, y), 54 p-stable, 159 p k q, 71 p ⊥ q, 143 p ⊥h q, 143 p  ϕ, 11 p  e, 121

167

168 pf , 3 p1 ⊗ · · · ⊗ pn , 73 p|B, 69 pf , 3 Pp , 124 Pα , 146 P≤α , 146 ϕ-formula, 11 generalized, 40 ϕ-type, 11 generalized, 40 ϕ −1 , 3 Ran (a/e), 147 RC, 94 RM, 96 R∞ , 94 stp(a/A), 59 SE (e), 104 Sϕ∗ (A), 40 S∆ (A), 19 Sϕ (A), 11 SU, 85, 115 SU(a/A), 86 SU(a/e), 115 tp∗ϕ (a/A), 40 T eq , 5 U, 85 U(a/A), 86 algebraic closure, 3 element, 3 type, 3 amalgamation base, 121 class, 124 analysability rank, 147 analysable, 147 analysis, 147 boolean space, 2 bounded closure, 102 hyperimaginary, 103 relation, 53 type, 59 c-free, 56 canonical base, 71, 124 canonical formula, 159 canonical parameter, 6 Cantor–Bendixson

Index degree, 8 rank, 8 closure algebraic, 102 bounded, 102 definable, 101 coheir, 43 coheir sequence, 45 complete type over a hyperimaginary, 104 continuous rank, 94 definable, 4 closure, 3 element, 3 type, 11 dependent theory, 18 diameter, 55 dividing, 21, 25 α times, 27 chain, 27 over a hyperimaginary, 110 Ehrenfeucht–Mostowski set, 33 elimination of hyperimaginaries, 131 equivalent hyperimaginaries, 101 finite equivalence relation, 6 relation, 53 finite character, 114 finitely satisfiable, 122 foreign, 144 forking, 25 extension, 25 over a hyperimaginary, 110 G-compactness, 66 heir, 43 hyperimaginary, 101 finitary, 101 length of, 101 countable, 101 imaginary, 5 independence of hyperimaginaries, 110 property, 15, 18 relation, 75 independence theorem, 65 for ^ | , 80 for hyperimaginary Lascar strong types, 119

169

Index over a model for hyperimaginaries, 115 independent, 31 index, 167–169 indiscernible sequence, 4 sequence of hyperimaginaries, 109 interbounded, 103 internal, 144 Kim–Pillay strong type, 59 Kim–Pillay topology, 138 Lascar inequalities, 88 rank, 85, 115 strong type, 53, 116 local D-rank, 19 local character, 31, 114 logic topology, 138 low formula, 136 theory, 136 Morley degree, 96 rank, 96 sequence, 32, 110 multiplicity, 69 nip, 18 nonforking extension, 25 independence, 31, 75 open mapping theorem, 51 order property, 13 orthogonal, 143 over h, 143 parallel types, 71 product of types, 73 properties of ^ | anti-reflexivity, 75 existence, 76 extension, 75 invariance, 75 left finite character, 75 left normality, 75

left transitivity, 75 local character, 75 monotonicity, 75 pairs lemma, 76 right base monotonicity, 75 right finite character, 80 right normality, 75 right transitivity, 80 strict, 75 symmetry, 77 rank abstract, 89 cantorian, 99 continuous, 96 real element, 5 simple theory, 18 small theory, 98 sort of a hyperimaginary, 101 splitting, 43 stable formula, 12 theory, 17 stationary type, 69, 121 strict order property, 15, 18 strong automorphism, 55, 116 heir, 45 type, 59 supersimple, 85 superstable, 85 thick formula, 54 totally transcendental, 97 tree property, 17 type hyperimaginary, 103 finitary, 1 global, 2 type-definable, 4 unstable formula, 12 theory, 17 weakly c-free, 56