Recent developments on absolute geometries and algebraization by K-loops
type: Item
instance of: scholarly article
title: Recent developments on absolute geometries and algebraization by K-loops
zbMATH Open document ID: 0941.51002
DOI: 10.1016/S0012-365X(99)00085-0
published in: Discrete Mathematics
publication date: 2000-07-30T00:00:00Z
review text: A spatial version of Pasch's assertion is obtained by starting from an ordered space \((P,{\mathcal L},\alpha),\) where \(P\) is the set of points, \({\mathcal L}\) the set of lines and \(\alpha \) the betweenness relation. It allows to define the concepts \(h\)-parallel, one sided h-parallel and hyperbolic incidence structure. A characterization of spaces with congruence \((P,{\mathcal L},\equiv)\) admitting point reflections is presented. An absolute space \((P,{\mathcal L},\alpha ,\equiv)\) is obtained by joining together incidence, congruence and order structures assumed to satisfy some compatibility conditions. A \(K\)-loop \((P,+)\) and an incidence fibration in the sense of \textit{E. Zizioli} [J. Geom. 30, 144-156 (1987; Zbl 0632.51019)] are associated to an absolute space satisfying certain conditions. They can be used for an algebraization of these spaces.
reviewed by: Nicolae Cotfas
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51F10
zbMATH DE Number: 1369794
zbMATH Keywords: loops
zbMATH Keywords: ordered spaces
zbMATH Keywords: incidence fibration
zbMATH Keywords: reflections
zbMATH Keywords: projective embedding
zbMATH Keywords: congruence
zbMATH Keywords: absolute space
author: Helmut Karzel
MaRDI profile type: Publication
Wikidata QID: Q126803871
full work available at URL: s0012-365x(99)00085-0
OpenAlex ID: W2084909919
description: scientific article; zbMATH DE number 1369794