Archimedisierung absoluter Ebenen. (Construction of archimedean absolute planes)
type: Item
instance of: scholarly article
title: Archimedisierung absoluter Ebenen. (Construction of archimedean absolute planes)
zbMATH Open document ID: 0576.51012
DOI: 10.1007/BF01222945
author: Detlef Gröger
published in: Journal of Geometry
publication date: 1985-01-01T00:00:00Z
review text: The author considers an ordered absolute plane in the sense of \textit{H. Karzel}, \textit{K. Sörensen} and \textit{D. Windelberg}: Einführung in die Geometrie (1973; Zbl 0248.50001). Depending on the choice of a unit length e one has a distance measure d with values in a commutative ordered group. Select a point 0. The set of points x with \(d(x,0)<n e\) for some \(n\in {\mathbb{N}}\) admits a partition into classes: two points x,y belong to the same class if \(d(x,y)<n^{-1}e\) for every \(n\in {\mathbb{N}}\). These classes become the points of a new plane whose structure stems from the original one. The new plane is not always an absolute plane. Examples are given. Remark. Similar questions without taking ordering into account were studied e.g. by \textit{A. Dress} in Math. Z. 85, 116-140 (1964; Zbl 0124.131) and Abh. Math. Semin. Univ. Hamb. 32, 52-54 (1963; Zbl 0159.051).
Mathematics Subject Classification ID: 51F05
Mathematics Subject Classification ID: 51G05
zbMATH DE Number: 3921373
zbMATH Keywords: archimedean absolute planes
MaRDI profile type: Publication
cites work: Q5533193
cites work: Zur Existenz von nichtkommutativen euklidischen Körpern
cites work: Q5662517
cites work: Q3881453
cites work: Euklidische Ebenen mit Nachbarelementen
cites work: Q5558376
cites work: Q5770489
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description: scientific article; zbMATH DE number 3921373