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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.

The subspace structure of Beidleman near-vector spaces is investigated. We characterise finite dimensional Beidleman near-vector spaces and we classify the R-subgroups of finite dimensional Beidleman near-vector spaces. We provide an algorithm to compute the smallest R-subgroup containing a given set of vectors. Finally, we classify the subspaces of finite dimensional Beidleman near-vector spaces.

A theory is developed in terms of which all finite-dimensional near-vector spaces over p (p a prime) are characterized.