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Preprint Number 782

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782. Dario Garcia, Dugald Macpherson, Charles Steinhorn
Pseudofinite structures and simplicity

Submission date: 30 September 2014.


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.

Mathematics Subject Classification: 03C13

Keywords and phrases:

Full text arXiv 1409.8635: pdf, ps.

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