Vectorspacelike representation of absolute planes
type: Item
instance of: scholarly article
title: Vectorspacelike representation of absolute planes
zbMATH Open document ID: 1117.51017
author: Mario Marchi
author: Helmut Karzel
published in: Journal of Geometry
publication date: 2007-06-07T00:00:00Z
review text: Let \((\mathbf{E},\mathcal{G},\alpha,\equiv)\) be an absolute plane, where \(\mathbf{E}\) and \(\mathcal{G}\) are the set of points and lines, respectively, \(\alpha\) is an order function, and \(\equiv\) is a congruence relation [see \textit{H. Karzel, K. Sörensen} and \textit{D. Windelberg}, Einführung in die Geometrie, Vandenhoeck, Göttingen (1973; Zbl 0248.50001)] with a multiplication defined on the points. The authors investigate a great number of properties of the plane. They show e.g., if \((\mathbf{E},\mathcal{G},\alpha,\equiv)\) is singular, then \((\mathbf{E},+)\) is a commutative group; also \((\mathbf{E},\mathcal{G},\alpha,\equiv)\) is vectorspacelike if it is Euclidean.
Mathematics Subject Classification ID: 51F05
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 5162056
zbMATH Keywords: absolute plane
zbMATH Keywords: vectorspacelike \(K\)-loop
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: s00022-006-1850-2
OpenAlex ID: W2017627054
DOI: 10.1007/S00022-006-1850-2
description: scientific article; zbMATH DE number 5162056