Archimedeisation of some ordered geometric structures which are related to kinematic spaces
type: Item
instance of: scholarly article
title: Archimedeisation of some ordered geometric structures which are related to kinematic spaces
zbMATH Open document ID: 0728.06013
DOI: 10.1007/BF03323288
author: C. J. Maxson
author: Helmut Karzel
published in: Results in Mathematics
publication date: 1991-01-01T00:00:00Z
review text: \textit{D. Gröger} has studied [J. Geom. 25, 49-76 (1985; Zbl 0576.51012); Mitt. Math. Ges. Hamb. 11, 441-457 (1987; Zbl 0632.51015)] the problem of `archimedeising' absolute planes, i.e. the conditions that a non- archimedean ordered absolute plane has to satisfy such that one can `extract' from it in a natural way an archimedean ordered absolute plane, once a `unit distance' has been selected. Given the close relationship between absolute planes and kinematic spaces (as defined by the first author [Sympos. Math. 11, Algebra commut., Geometria, Convegni 1971/1972, 413-439 (1973; Zbl 0284.50017)]), the authors aim to `archimedeise' ordered kinematic spaces and lay, in this paper, the algebraic groundwork for this task. It leads through the study of linearly ordered groups, cyclic ordered groups, betweenness groups, and separated groups, to the definition of a notion of `archimedeanity' and `archimedeisation' for separated groups, and eventually to the complete discussion of the archimedeisation of three (out of four possible) types of separated groups that are derived from unitary associative algebras (A,K) of degree 2 over non-archimedean ordered fields K. The fourth type will be treated in another paper.
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 51J15
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 20F60
zbMATH DE Number: 4202330
zbMATH Keywords: archimedeization of separated groups
zbMATH Keywords: absolute planes
zbMATH Keywords: ordered kinematic spaces
reviewed by: Victor V. Pambuccian
MaRDI profile type: Publication
cites work: Q3241211
cites work: Q5674022
cites work: Archimedisierung absoluter Ebenen. (Construction of archimedean absolute planes)
cites work: Q4771790
cites work: Kinematische Algebren und ihre geometrischen Ableitungen
cites work: Q3863930
cites work: Q3998173
cites work: Q5662517
full work available at URL: https://doi.org/10.1007/bf03323288
OpenAlex ID: W2044793631
description: scientific article; zbMATH DE number 4202330