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Finite Structures with Few Types. (AM-152), Volume 152 : Annals of Mathematics Studies - Gregory Cherlin

Finite Structures with Few Types. (AM-152), Volume 152

Annals of Mathematics Studies


Published: 12th January 2003
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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.

Introductionp. 1
Basic Notionsp. 11
Finiteness Propertiesp. 11
Rankp. 18
Imaginary Elementsp. 23
Orthogonalityp. 31
Canonical Projective Geometriesp. 36
Smooth Approximabilityp. 40
Envelopesp. 40
Homogeneityp. 42
Finite Structuresp. 46
Orthogonality Revisitedp. 50
Lie Coordinatizationp. 54
Finiteness Theoremsp. 63
Geometrical Finitenessp. 63
Sectionsp. 67
Finite Languagep. 71
Quasifinite Axiomatizabilityp. 75
Ziegler's Finiteness Conjecturep. 79
Geometric Stability Generalizedp. 82
Type amalgamationp. 82
The sizes of envelopesp. 90
Nonmultidimensional Expansionsp. 94
Canonical Basesp. 97
Modularityp. 101
Local Characterization of Modularityp. 104
Reducts of Modular Structuresp. 107
Definable Groupsp. 110
Generation and Stabilizersp. 110
Modular Groupsp. 114
Dualityp. 120
Rank and Measurep. 124
The Semi-Dual Coverp. 126
The Finite Basis Propertyp. 134
Reductsp. 141
Recognizing Geometriesp. 141
Forgetting Constantsp. 149
Degenerate Geometriesp. 153
Reducts with Groupsp. 156
Reductsp. 164
Effectivityp. 170
The Homogeneous Casep. 170
Effectivityp. 173
Dimension Quantifiersp. 178
Recapitulation and Further Remarksp. 183
Referencesp. 187
Indexp. 191
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691113326
ISBN-10: 0691113327
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 200
Published: 12th January 2003
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.91
Weight (kg): 0.31