The model theory of Commutative Near Vector Spaces

TL;DR

This paper explores the model theory of commutative near vector spaces, revealing conditions under which they are vector spaces, and establishing axiomatizability and quantifier elimination for finite block cases.

Contribution

It demonstrates that regular near vector spaces are actually vector spaces and establishes axiomatizability and quantifier elimination for finite block near vector spaces.

Findings

Regular near vector spaces are vector spaces.

Finite block near vector spaces are axiomatizable.

Quantifier elimination holds for finite block near vector spaces.

Abstract

In this paper we study near vector spaces over a commutative $F$ from a model theoretic point of view. In this context we show regular near vector spaces are in fact vector spaces. We find that near vector spaces are not first order axiomatisable, but that finite block near vector spaces are. In the latter case we establish quantifier elimination, and that the theory is controlled by which elements of the pointwise additive closure of $F$ are automorphisms of the near vector space.