Dimension, matroids, and dense pairs of first-order structures

Details Dimension, matroids, and dense pairs of first-order structures Dimension, matroids, and dense pairs of first-order structures

2009-07-24

arxiv.org/abs/0907.4237v2

Annals of Pure and Applied Logic Volume 162, Issue 7, June-July 2011, Pages 514-543

Mathematics

Logic

Version 2.8. 61 pages

Scientific paper

10.1016/j.apal.2011.01.003

A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of omega and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique existential matroid. Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding a field and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We extend the above result to dense tuples of structures.

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