Pseudofinite structures and simplicity

Journal of Mathematical Logic 15 (1):1550002 (2015)
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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.

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Citations of this work

Model theory of finite and pseudofinite groups.Dugald Macpherson - 2018 - Archive for Mathematical Logic 57 (1-2):159-184.
Dividing and weak quasi-dimensions in arbitrary theories.Isaac Goldbring & Henry Towsner - 2015 - Archive for Mathematical Logic 54 (7-8):915-920.
Pseudofinite difference fields.Tingxiang Zou - 2019 - Journal of Mathematical Logic 19 (2):1950011.
Pseudofinite difference fields and counting dimensions.Tingxiang Zou - 2020 - Journal of Mathematical Logic 21 (1):2050022.

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References found in this work

Simple theories.Byunghan Kim & Anand Pillay - 1997 - Annals of Pure and Applied Logic 88 (2-3):149-164.
On Pseudo-Finite Dimensions.Ehud Hrushovski - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):463-495.
From stability to simplicity.Byunghan Kim & Anand Pillay - 1998 - Bulletin of Symbolic Logic 4 (1):17-36.
Asymptotic Classes of Finite Structures.Richard Elwes - 2007 - Journal of Symbolic Logic 72 (2):418 - 438.
Definability and definable groups in simple theories.Anand Pillay - 1998 - Journal of Symbolic Logic 63 (3):788-796.

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