Quasiparallel planes in finite planar spaces
The paper under review contains characterizations of finite generalized three-dimensional projective spaces and three-dimensional affine spaces based on quasiparallelism of subspaces. Let \({\mathbb P} =({\mathcal S}, {\mathcal L}, {\mathcal P})\) be a finite planar space, and let \(X\) and \(X'\) be two subspaces of \({\mathbb P}\). Then \(X\) and \(X'\) are quasiparallel if \(|X \cap \ell |= |X' \cap \ell|\) for all lines \(\ell\not\subseteq X\cup X'\) [cf. \textit{N. Melone} and \textit{U. Ott}, Des. Codes Cryptography 2, No. 4, 307-313 (1992; Zbl 0763.05014)]. The author generalizes to the planar case her investigations on quasiparallelism in finite linear spaces [cf. \textit{S. Freni}, Ric. Mat. 47, No. 1, 115-124 (1998; Zbl 0926.05009); Discrete Math. 208-209, 273-284 (1999; Zbl 0943.51001)]. In the paper under review the following two results are proved. Theorem 1. A finite planar space \(\mathbb P\) is a generalized projective space if, and only if, planes in \(\mathbb P\) are quasiparallel. Theorem 2. Let \(\mathbb P\) be a finite linear space with the following two properties. (i) There are no two planes \(\alpha\) and \(\beta\) such that \({\mathcal S} = \alpha \cup\beta\). (ii) There is a line with at least four points. Then \(\mathbb P\) is an affine space of order \(n\geq 4\) if, and only if, for each plane \(\alpha\) there is a plane \(\alpha'\) quasiparallel to \(\alpha\) such that \(\alpha\cap \alpha' = \emptyset\).
scientific article; zbMATH DE number 1838885