Zur Existenz und Fortsetzbarkeit von o-Epimorphismen angeordneter projektiver Ebenen. (On the existence and extension of o-epimorphisms of ordered projective planes.)
type: Item
instance of: scholarly article
title: Zur Existenz und Fortsetzbarkeit von o-Epimorphismen angeordneter projektiver Ebenen. (On the existence and extension of o-epimorphisms of ordered projective planes.)
zbMATH Open document ID: 0686.51005
DOI: 10.1007/BF02941375
author: Peter Hartmann
author: Sibylla Priess-Crampe
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1988-01-01T00:00:00Z
review text: Let \(\phi\) be an order-preserving epimorphisms of a projective plane \({\mathfrak F}\) lying in an ordered projective plane \({\mathfrak E}\). Let U, V be points and a, b, c, d, w lines of \({\mathfrak F}\) with \(U^{\phi}\neq V^{\phi}\), \(U\in a,b,w\) and \(V\in c,d,w\). The set \({\mathcal V}=(a,b,c,d)\) of all points \(P\in {\mathfrak E}\), where a and b separate P and V, and c and d separate P and U, is called a quadrangle with respect to U and V. \({\mathcal V}\) is called admissible for a point \(P\in {\mathcal E}\) if there exists a quadrangle \({\mathcal V}'=(a',b',c',d')\) with respect to U and V such that \(P\in {\mathcal V}'\subset {\mathcal V}\) and the images of a, b, c, d, a', b', c', d' and w under \(\phi\) are pairwise distinct. Let \({\mathcal V}_{U,V}(P)\) denote the set of quadrangles with respect to U and V that are admissible for P. The authors prove that \(\phi\) can be extended to an order-preserving epimorphism of \({\mathfrak E}\), if the `Vierecksbedingung' is fulfilled, i.e., if for all points of P of \({\mathfrak E}\), U,V,W of \({\mathfrak F}\) with \({\mathcal V}_{U,W}(P)\neq \emptyset\), and for all \({\mathcal V}\in {\mathcal V}_{U,V}(P)\) there exists a quadrangle \({\mathcal W}\in {\mathcal V}_{U,W}(P)\) such that \({\mathcal W}\subset {\mathcal V}.\) Although, even in the desarguesian case, this condition is not necessary, the paper under consideration provides us with strong and deep results on the existence of order-preserving epimorphisms in arbitrary projective planes: In a second part of their work, the authors show that the `Vierecksbedingung' is satified whenever \({\mathfrak F}\) is order-dense in \({\mathfrak E}\) or whenever \(\phi\) (\({\mathfrak F})\) is an Archimedean ordered plane. The latter allows for instance, to carry over the classical construction of natural places (valuations) of ordered fields to all planes admitting an Archimedean subplane, and thus to all ordered translation planes.
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 51A10
zbMATH DE Number: 4124472
zbMATH Keywords: epimorphisms of projective planes
zbMATH Keywords: order compatible homomorphisms
zbMATH Keywords: natural epimorphism
zbMATH Keywords: ordered projective plane
reviewed by: Franz B. Kalhoff
MaRDI profile type: Publication
cites work: Zur Fortsetzung von o-Stellen angeordneter Divisionsalgebren. (On extensions of o-places of ordered division algebras)
cites work: Zur Archimedisierung projektiver Ebenen. (Archimedean projective planes)
cites work: Allgemeine Bewertungstheorie.
cites work: Q3223453
cites work: o-Epimorphismen angeordneter projektiver Ebenen. (o-epimorphisms of ordered projective planes)
cites work: A homomorphism theorem for projective planes
description: scientific article; zbMATH DE number 4124472