Finite Structures with Few Types

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Princeton University Press, 2003 - Mathematics - 193 pages
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This book applies model theoretic methods to the study of certain finite permutation groups, the automorphism groups of structures for a fixed finite language with a bounded number of orbits on 4-tuples. Primitive permutation groups of this type have been classified by Kantor, Liebeck, and Macpherson, using the classification of the finite simple groups.


Building on this work, Gregory Cherlin and Ehud Hrushovski here treat the general case by developing analogs of the model theoretic methods of geometric stability theory. The work lies at the juncture of permutation group theory, model theory, classical geometries, and combinatorics.


The principal results are finite theorems, an associated analysis of computational issues, and an "intrinsic" characterization of the permutation groups (or finite structures) under consideration. The main finiteness theorem shows that the structures under consideration fall naturally into finitely many families, with each family parametrized by finitely many numerical invariants (dimensions of associated coordinating geometries).


The authors provide a case study in the extension of methods of stable model theory to a nonstable context, related to work on Shelah's "simple theories." They also generalize Lachlan's results on stable homogeneous structures for finite relational languages, solving problems of effectivity left open by that case. Their methods involve the analysis of groups interpretable in these structures, an analog of Zilber's envelopes, and the combinatorics of the underlying geometries. Taking geometric stability theory into new territory, this book is for mathematicians interested in model theory and group theory.

 

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Contents

Introduction
1
12 Results
4
Basic Notions
11
22 Rank
18
23 Imaginary Elements
23
24 Orthogonality
31
25 Canonical Projective Geometries
36
Smooth Approximability
40
55 Modularity
101
56 Local Characterization of Modularity
104
57 Reducts of Modular Structures
107
Definable Groups
110
62 Modular Groups
114
63 Duality
120
64 Rank and Measure
124
65 The SemiDual Cover
126

32 Homogeneity
42
33 Finite Structures
46
34 Orthogonality Revisted
50
35 Lie Coordinatization
54
Finiteness Theorms
63
42 Sections
67
43 Finite Language
71
44 Quasifinite Axiomatizability
75
45 Zieglers Finiteness Conjecture
79
Geometric Stability Generalized
82
52 The sizes of envelopes
90
53 Nonmultidimensional Expansions
94
54 Canonical Bases
97
66 The Finite Basis Property
134
Reducts
141
72 Forgetting Constants
149
73 Degenerate Geometries
153
74 Reducts with Groups
156
75 Reducts
164
Effectivity
170
82 Effectivity
173
83 Dimension Quantifiers
178
84 Recapitulation and Further Remarks
183
REFERENCES
187
INDEX
191
Copyright

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Common terms and phrases

Popular passages

Page 187 - P. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc., 13(1981), 1-22.
Page 187 - Fried and M. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 1 1 (Springer, New York, 1986).
Page 187 - Dordrecht, 1997. [CHL] G. Cherlin, L. Harrington, and A. Lachlan, No-categorical, No-stable structures, Annals of Pure and Applied Logic 18 (1980), 227-270.

About the author (2003)

Gregory Cherlin is Professor of Mathematics at Rutgers University. He is the author of Model Theoretic Algebra: Selected Topics. Ehud Hrushovski is Professor of Mathematics at the Hebrew University of Jerusalem.

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