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Preprint Number 195
195. Antongiulio Fornasiero
Dimension, matroids, and dense pairs of first-order structures
Submission date: 24 July 2009. Revised 26 February 2010.
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.
Mathematics Subject Classification: Primary: 03Cxx; Secondary: 03C64
Keywords and phrases: Topological structures, pregeometries, existential matroids, dimension, dense pairs, d-minimal
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