On hyperplanes and free subspaces of affine Klingenberg spaces

Summary

In a previous paper we introduced the notion of an affine Klingenberg space and described two equivalent axiom systems. The main tool for establishing this equivalence was the internal description of finite dimensional free subspaces. Our present axiom system yields the existence of maximal independent subsets and so we can introduce the notion of a hyperplane as a (free) subspace generated by a maximal independent subset with one point removed. However, we are not able to establish elementary properties of hyperplanes, analogous to those found in the ordinary case, within the confines of our existing axiom system. In this paper we introduce two more axioms that allow us to characterize free subspaces of arbitrary dimension and, in particular, the freeness of our space in terms of geometric properties of hyperplanes. However, we have still to establish when our space is desarguesian, when it can be coordinatized by some module over a local ring and when it can be embedded in a projected Klingenberg space.