The model theory of commutative 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.