Table Of ContentSimple Theories
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science. Amsterdam, The Netherlands
Volume 503
Simple Theories
by
Frank 0. Wagner
lnstitut Girard Desargues,
Universite Claude Bernard (Lyon-I),
Villeurbanne, France
....
''
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5417-3 ISBN 978-94-017-3002-0 (eBook)
DOI 10.1007/978-94-017-3002-0
Printed on acid-free paper
All Rights Reserved
© 2000 Springer Science+B usiness Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
Softcover reprint of the hardcover 1st edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
FOr meine Eltern
Contents
Preface lX
Acknowledgements Xl
1. PRELIMINARIES 1
1.1 Introduction 1
1.2 Notation and model-theoretic prerequisites 5
1.3 Examples 10
1.4 Bibliographical remarks 13
2. SIMPLICITY 15
2.1 The monster model and imaginaries 15
2.2 Dividing and forking 18
2.3 Simplicity 22
2.4 Morley sequences 27
2.5 The Independence Theorem 30
2.6 Simplicity and independence 36
2.7 Bounded equivalence relations 38
2.8 Types 41
2.9 Stability 47
2.10 Bibliographical remarks 50
3. HYPERIMAGINARIES 51
3.1 Hyperimaginaries 51
3.2 Forking for hyperimaginaries 57
3.3 Canonical bases 62
3.4 Internality and analysability 68
3.5 ?-closure and local modularity 74
3.6 Elimination of hyperimaginaries 78
3.7 The Lascar group 84
3.8 Bibliographical remarks 93
vii
viii SIMPLE THEORIES
4. GROUPS 95
4.1 Type-definable groups 95
4.2 Relatively definable groups 105
4.3 Hyperdefinable groups 110
4.4 Chain conditions and commensurativity 116
4.5 Stabilizers 121
4.6 Quotient groups and analysability 129
4.7 Generically given groups 133
4.8 Locally modular groups 138
4.9 Bibliographical remarks 145
5. SUPERSIMPLE THEORIES 147
5.1 Ranks 147
5.2 Weight and domination 155
5.3 Elimination of hyperimaginaries 161
5.4 Supersimple groups 167
5.5 Type-definable supersimple groups 173
5.6 Supersimple division rings 178
5.7 Bibliographical remarks 186
6. MISCELLANEOUS 187
6.1 Small theories 187
6.1.1 Elimination of hyperimaginaries 189
6.1.2 Locally modular theories 193
6.1.3 Theories with finite coding 195
6.1.4 Lachlan's Conjecture 202
6.2 w-categorical theories 204
6.2.1 An amalgamation construction 205
6.2.2 w-categorical supersimple groups 213
6.2.3 w-categorical CM-trivial theories 221
6.3 Simple expansions of simple theories 225
6.3.1 Amalgamating simple theories 225
6.3.2 Simple theories with an automorphism 233
6.4 Low theories 239
6.5 Bibliographical remarks 242
Bibliography 245
Index 257
Preface
The class of simple theories extends that of stable theories, and contains some
important structures, such as the random graph, pseudo-finite fields, and fields
with a generic automorphism. Following Kim's proof that Shelah's notion of
''forking independence" is symmetric in a simple theory, this area of model
theory has been a field of intense study, placing stability theory into a wider
framework, and thus serving to explain a phenomenon previously noticed in
connection with the model theory of pseudo-finite fields, namely the appearance
of stability-theoretic behaviour in an unstable context.
The generalization has required some important new tools, in particular the
model-theoretic treatment of hyperimaginaries (classes modulo type-definable
equivalence relations). While many of the results known for stable theories
have been generalized to simple structures, some fundamental results from
stability theory are as yet only conjectural for the simple case (e.g. the group
configuration theorem, or the binding group theorem).
In this book, I shall present the (or rather: my) present knowledge of sim
plicity theory, viewing it not as an appendix to stability theory, but as a general
theory of which the stable structures form a particular case. It is intended both
as an introduction for the graduate student, and as a reference for research in
this area.
ix
Acknowledgements
Parts of this book were written while I held a Heisenberg-Stipendium of the
Deutsche Forschungsgemeinschaft (Wa 899/2-1) at the University of Oxford.
I should like to thank the DFG for its financial support, the Mathematical
Institute for its hospitality, and St. Catherine's College for looking after me so
well. Other parts were written during various sojourns in Tokyo; I am grateful to
the Kanto Model Theory group and Waseda University for the friendly welcome
extended to me. Finally, the manuscript was finished at the Universite Claude
Bernard at Lyon, where again I was received warmly. Further thanks are due to
the Fields Institute for Research in the Mathematical Sciences at Toronto and
to the Mathematical Sciences Research Institute at Berkeley, both of which I
had the opportunity to visit for extended periods, and where many of the results
in this book originated.
I am indebted to Angus Macintyre for starting it all off, to Anand Pillay
for sharing his extensive knowledge and insight, and to Ambar Chowdhury,
Bradd Hart, and Byunghan Kim for many discussions on simplicity theory.
Thanks also to Steven Buechler, Enrique Casanovas, Zoe Chatzidakis, David
Evans, Ehud Hrushovski, Masanori Itai, Hirotaka Kikyo, Daniel Lascar, Chris
Laskowski, Dugald McPherson, Keishi Okamoto, Bruno Poizat, Akita Tsuboi,
Martin Ziegler, and Boris Zil'ber, all of whom contributed in some form or
other.
P.O.W.
Xl
