On hyperplanes and free subspaces of affine Klingenberg spaces
Previously [in Combinatorics, Ravello 1988, 185-200, Rende (1991)], the authors described two equivalent axiom systems for affine Klingenberg spaces \({\mathcal A} = (P, {\mathcal L}, \|, \sim)\). In the present paper, they study one of these systems in more detail. A maximal independent subset of \({\mathcal A}\) maps to a maximal independent subset of the ordinary affine space \({\mathcal A}^* = {\mathcal A}/ \sim\). A subspace \({\mathcal S}\) of \({\mathcal A}\) is said to be free if \({\mathcal S}\) is spanned by an independent set of points. It follows that \(\dim {\mathcal S} = \dim\;{\mathcal S}^*\). A subspace \({\mathcal H}\) is called a hyperplane if \({\mathcal H} = \langle M \backslash \{h\} \rangle\) for some independent point set \(M\) and \(h \in M\). Equivalently, \({\mathcal H}\) is free and \({\mathcal H}^*\) is a hyperplane of \({\mathcal A}^*\). In order to obtain basic properties of hyperplanes , two additional axioms are introduced; one requires that a hyperplane and a non-neighbouring line meet in at most 1 point. Freeness of \({\mathcal A}\) can then be characterized in terms of geometric properties of hyperplanes, but there remain several open questions.
scientific article; zbMATH DE number 699772