Partial affine spaces of dimension \(\geq 3\)
An incidence structure \((P, {\mathcal G})\) is called a partial linear space if two points are on at most one line and, dually, two lines have at most one point in common. A partial linear space with parallelism is called a partial affine space if there exists an affine space \((P,{\mathcal G}')\) such that \({\mathcal G} \subseteq {\mathcal G}'\) and such that the parallel relation on \(\mathcal G\) is induced from the natural parallelism in \({\mathcal G}'\). Let \((P,{\mathcal G})\) be a partial linear space with parallelism such that the Tamaschke configuration is closed. If certain conditions are fulfilled, which make sure that there are ``enough'' connectible points in \(P\), and that the dimension is at least three, then \((P,{\mathcal G})\) is a partial affine space. The theorem has been proved by \textit{A. Herzer} and the author [J. Geom. 50, No. 1-2, 124-142 (1994; Zbl 0807.51002)], and improved by \textit{V. Pambuccian} [J. Geom 55, No. 1-2, 139-140 (1996)], under the assumption that the dimension is at least four.
scientific article; zbMATH DE number 913040