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

Abstract

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.

Mathematics Subject Classification (2020):

• 51B05
• 16Y30
• 12K05
• 05B35

Key words:

• Nearaffine spaces
• near-linear spaces
• near-vector spaces
• nearfields
• incidence geometry