%0 Journal Article
%T Dimension, matroids, and dense pairs of first-order structures
%A Antongiulio Fornasiero
%J Mathematics
%D 2009
%I arXiv
%R 10.1016/j.apal.2011.01.003
%X 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.
%U http://arxiv.org/abs/0907.4237v2