Nearaffine spaces and their application to near-vector spaces

The nearaffine space, defined by J. Andr ́e, is a generalisation of the traditional affine space, for which the line joining two points generally depends on the order in which the points are joined, i.e. the map sending a pair of different points to their commmon line is noncommutative. This geometry is strongly linked to Andr ́e’s near-vector space, and was recently used to solve an open problem in the study of near-vector spaces, namely that a subset of a near-vector space that is closed under addition and scalar multiplication is a subspace. Furthermore, an analogue for the projective space of a vector space is developed for near-vector spaces, using the geometry of a nearaffine space.