Vector spaces with a union of independent subspaces

TL;DR

This paper investigates the model theory of vector spaces over infinite and finite fields with a predicate for a union of independent subspaces, establishing completeness, quantifier elimination, and near-model-completeness results.

Contribution

It introduces a new framework for analyzing vector spaces with union predicates, proving completeness and quantifier elimination for infinite fields and near-model-completeness for finite fields.

Findings

Complete theory for infinite fields with union of independent subspaces

Quantifier elimination in the expanded language for infinite fields

Near-model-completeness for finite fields with the union predicate

Abstract

Motivated by the theory of locally definable groups, we study the theory of $K$-vector spaces with a predicate for the union $X$ of an infinite family of independent subspaces. We show that if $K$ is infinite then the theory is complete and admits quantifier elimination in the language of $K$-vector spaces with predicates for the $n$-fold sums of $X$ with itself. If $K$ is finite this is no longer true, but we still have that a natural completion is near-model-complete.