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

TL;DR

This paper introduces the concept of structures with an existential matroid, generalizing pregeometric structures, and explores their properties, including dense pairs and tuples, with applications to fields and superstable groups.

Contribution

It generalizes pregeometric structures to include existential matroids and extends results on dense pairs and tuples in structures expanding fields.

Findings

Ultraproducts of pregeometric structures have a unique existential matroid.

Theories of dense elementary pairs have natural completions with models also possessing a unique existential matroid.

Extensions to dense tuples of structures expanding fields are established.

Abstract

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.