On independence in near-vector spaces and their matroids

Abstract

In this paper we study and characterise independence in near-vector spaces constructed using copies of finite fields. We show that for regular near-vector spaces of this nature, independence is equivalent to the notion of independence in the associated vector space. We prove that for the construction where the number of maximal regular subspaces coincides with the dimension, there are more elements outside of the quasi-kernel which can generate the entire space than previously thought. As a highlight, we completely characterise independence for this space and define matroids for finite field constructions and those using copies of a proper finite near-field.