Finite Structures with Few Types

Gregory Cherlin, Ehud Hrushovski

194 pages, parution le 21/02/2003

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Résumé

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.

Contents

Introduction
Basic Notions
Smooth Approximability
Finiteness Theorems
Geometry Stability Generalized
Definable Groups
Reducts
Effectivity

L'auteur - Gregory Cherlin

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

L'auteur - Ehud Hrushovski

Ehud Hrushovski is Professor of Mathematics at the Hebrew University of Jerusalem.

Caractéristiques techniques

	PAPIER
Éditeur(s)	Princeton University Press
Auteur(s)	Gregory Cherlin, Ehud Hrushovski
Parution	21/02/2003
Nb. de pages	194
Format	15,5 x 23,5
Couverture	Broché
Poids	316g
Intérieur	Noir et Blanc
EAN13	9780691113326
ISBN13	978-0-691-11332-6