Convexity in halfordered planes
Let \((E, {\mathcal G}, \alpha)\) be a halfordered plane, i.e. a linear space \((E, {\mathcal G})\) with at least three points on each line endowed with a mapping \(\alpha : \{(a,b,c) \in E^3 \mid a \neq b,c\) and \(a,b,c\) collinear\} \(\to \{\pm 1\}\) such that the Pasch axiom and the cancellation rule \(\alpha (a,b,c) \cdot \alpha (a,c,d) = \alpha (a,b,d)\) hold (read: `\(a\) lies between \(b\) and \(c\)', if and only if \(\alpha (a,b,c) = - 1)\). Then \(\alpha\) induces a convexity on \(E\) (i.e. a system \(K_\alpha\) of subsets of \(E\) closed under intersection and containing \(\emptyset\), \(E\) and all lines) by calling \(M \subset E\) convex, if \(a,b \in M\) implies that all points lying between \(a\) and \(b\) are in \(M\). The convexities arising in this way are not strong enough, to properly describe the underlying halforderings, for in general, the segments \([a,b] : = \{a,b\} \cup \{x \in E \mid \alpha (x,a,b) = - 1\}\) are not convex. The authors present some examples and prove the equivalence of the following four statements: (1) each segment \([a,b]\) lies in \(K_\alpha\), (2) each segment \([a,b]\) is the convex hull of \(\{a,b\}\), (3) for mutually distinct \(a,b,x,p \in E\) one has that \(\alpha (x,a,b) = \alpha (p,x,a) = - 1\) implies \((\alpha (p,a,b) = - 1\), (4) \(\alpha\) is an ordering (i.e. fulfills the rule that for mutually distinct collinear points \(a,b,c\) exactly one of the values \(\alpha (a,b,c)\), \(\alpha (b,a,c)\), \(\alpha (c,a,b)\) equals \(- 1)\), or \(\alpha\) is nearly trivial (meaning in particular that \(\alpha (a,b,c)\) depends only on \(b\) and \(c)\), or \((E, {\mathcal G})\) is the affine plane over \(GF(3)\).
scientific article; zbMATH DE number 827390