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Pseudofinite structures and simplicity

Darío García

Departamento de Matemáticas, Universidad de los Andes, Cra 1 No. 18A-10, Edificio H, Bogotá 111711, Colombia

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Dugald Macpherson

School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

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, and

Charles Steinhorn

Vassar College, Poughkeepsie, NY 12604, USA

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https://doi.org/10.1142/S0219061315500026Cited by:10 (Source: Crossref)

Abstract

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.

Keywords:

AMSC: 03C13, 03C45, 03C52, 03C98

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