Ordered symmetric Minkowski planes I
type: Item
instance of: scholarly article
title: Ordered symmetric Minkowski planes I
zbMATH Open document ID: 1206.51004
author: Helmut Karzel
author: Jarosław Kosiorek
author: Andrzej Matraś
published in: Journal of Geometry
publication date: 2009-11-23T00:00:00Z
review text: Let \({\mathcal{M}}\) be a Minkowski plane. It is well known [see \textit{R. Artzy}, J. Geometry 3, 93--102 (1973; Zbl 0245.50031), \textit{W. Benz}, Vorlesungen über Geometrie der Algebren. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0258.50024), \textit{H. Karzel}, Math. Z. 62, 268--291 (1955; Zbl 0066.38803)], that if the rectangle axiom holds then an orthogonality between circles can be defined and there exists a commutative field \(F\) such that the circles of \({\mathcal{M}}\) can be identified with the elements of the projective linear group \(PGL(2, F)\). In particular two circles \(A, B\) are orthogonal (\(A {\perp}B\)) iff \(A \neq B\) and \(A ^{ - 1} B = B ^{ - 1} A\). In the paper the case \(char F \neq 2\) is considered and the authors introduce a ``valuation'' of the circles, namely a function \(h\) that associates to each circle one of the values \(1\) or \(-1\). The main condition for the valuation \(h\) demands that for three circles \(A,B,C\), where \(A\) is orthogonal to \(B\) and \(C\) and the circle \(D= B \cdot \,A^{-1} \cdot C\) the product \(h(A)\cdot h(B)\cdot h(C) \cdot h(D)\) equals \(1\). If this condition is valid the pair \(({\mathcal{M}},h)\) is called a halfordered symmetric Minkowski plane and \(h\) an orthogonal valuation. In the paper the authors prove that there is a one to one correspondence between halforders of the field \(F\) and orthogonal valuation of symmetric Minkowski planes. In a forthcoming paper the notion of ``separation'' for quadruples of concyclic points will be derived from a valuation and so the connection with the concept of an ordered symmetric Minkowski plane given in [\textit{H. J. Kroll}, Abh. Math. Semin. Univ. Hamb. 46, 217--255 (1977; Zbl 0374.50006)] will be established.
reviewed by: Gloria Rinaldi
Mathematics Subject Classification ID: 51B20
Mathematics Subject Classification ID: 51G05
zbMATH DE Number: 5637106
zbMATH Keywords: Minkowski plane
zbMATH Keywords: order
zbMATH Keywords: valuation
MaRDI profile type: Publication
full work available at URL: s00022-009-1896-z
OpenAlex ID: W2014583625
DOI: 10.1007/S00022-009-1896-Z
description: scientific article; zbMATH DE number 5637106