Subspace operations in affine Klingenberg spaces

In two previous papers we introduced the notion of an Ane Klingenberg space A and presented a geometric description of its free subspaces. Presently, we consider the operations of join, intersection and parallelism on the free subspaces ofA: As in the case of ordinary ane spaces, we obtain the Parallel Postulate. The situation with join and intersection is not that straightforward. In particular, the central problem is whether the join of two free subspaces is free? We show that if A is not an ordinary ane space and dim A 4t henA has a subspace which is both not free and the join of two free subspaces. Thus, join and intersection do not possess the usual closure properties. We determine necessary and sucient conditions under which the join of two free subspaces is free, and in such a case we verify the Dimension Formula. The subspace operations are essential tools for establishing whenA is desarguesian and when it can be embedded in a projective Klingenberg space.