Theorie der verallgemeinerten kinematischen Räume. (Theory of generalized kinematic spaces)
type: Item
instance of: scholarly article
title: Theorie der verallgemeinerten kinematischen Räume. (Theory of generalized kinematic spaces)
zbMATH Open document ID: 0636.51012
DOI: 10.1007/BF03322398
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Incidence spaces (G,\({\mathfrak G})\) are considered with regular sets of collineations, the so-called incidence loops. To any metric plane there belongs such a structure namely a kinematic space which is embedded into a projective space P such that the complement Q is a 2-set: \(Q\subset P\) is called a 2-set if for any line \(X\subset P\) one has \(| X\cap Q| \leq 2\) or \(X\subset Q\). In the first part of the paper a survey is given on those 2-sets, especially on s-homogeneous ones. Q is called regular, if \(P\setminus Q\) is an incidence loop. It is shown that knot ovals Q with \(| Q| \neq 4\) are not regular and that an ovoid in a 3-dimensional projective space can not be a 2-set of an incidence group. So, there is the existence problem: Does any regular Q contain at least one line? In the second part properties of incidence groups are generalized to incidence loops. Of special interest are the Moufang incidence loops with parallelism. Furthermore it is shown that any linear fibered Moufang incidence loop with a 2-set as a complement in a projective space is two- sided, and a classification of all those incidence loops is given.
Mathematics Subject Classification ID: 51J05
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 51A15
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 4036595
zbMATH Keywords: metric plane
zbMATH Keywords: kinematic space
zbMATH Keywords: 2-set
zbMATH Keywords: incidence loop
zbMATH Keywords: ovoid
zbMATH Keywords: incidence groups
zbMATH Keywords: Moufang incidence loops with parallelism
author: Günter Peter Kist
reviewed by: Herbert Hotje
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4036595