Geometries with non-commutative joins and their application to near-vector spaces

In this paper we add to the theory of the geometry of near-structures. More specifically, we define a near-linear space, prove some properties and show that by adding some axioms we arrive at a nearaffine space, as defined by André. As a highlight, we use some of the geometric results to prove an open problem in near-vector space theory, namely that a subset of a near-vector space that is closed under addition and scalar multiplication is a subspace. We end the paper with a first look at the projections of nearaffine spaces.