Convexity in halfordered planes
type: Item
instance of: scholarly article
title: Convexity in halfordered planes
zbMATH Open document ID: 0840.51008
DOI: 10.1007/BF01222853
author: Renate Jaritz
author: Alexander Kreuzer
published in: Journal of Geometry
publication date: 1996-02-01T00:00:00Z
review text: 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 \(GF(3)\).
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 827390
zbMATH Keywords: halfordered planes
zbMATH Keywords: ordered planes
zbMATH Keywords: convexity
zbMATH Keywords: abstract convexity
reviewed by: Franz B. Kalhoff
MaRDI profile type: Publication
cites work: Q5691389
cites work: Konvexität in halbgeordneten projektiven und affinen Räumen
cites work: Zur Abhängigkeit von Anordnungsaxiomen. (On relations between axioms of betweenness)
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cites work: Halbgeordnete Ebenen. (On halfordered planes)
cites work: Die Ordnungsfunktionen einer Geometrie
cites work: Q5794097
cites work: Konvexität bei Ordnungsfunktionen
description: scientific article; zbMATH DE number 827390