Über pseudoaffine Räume. (On pseudoaffine spaces)
type: Item
instance of: scholarly article
title: Über pseudoaffine Räume. (On pseudoaffine spaces)
zbMATH Open document ID: 0639.51003
DOI: 10.1007/BF01222397
published in: Journal of Geometry
publication date: 1988-01-01T00:00:00Z
review text: A partial line space all of whose subplanes are affine is called pseudoaffine. A pseudoaffine space must be affine if its order exceeds three. The author treats pseudoaffine spaces \((P,{\mathcal L})\) of order 3. Such spaces are shown to correspond biuniquely to commutative Moufang loops of exponent 3. Using this method of coordinatization the author solves three problems which were posed by H. Karzel [cf. \textit{H. Karzel} and \textit{I. Pieper}, Jahresber. Dtsch. Math.-Ver. 72, 70-114 (1970; Zbl 0202.510), p. 85, and \textit{H. Karzel}, Mitt. Math. Ges. Hamb. 11, 355-368 (1985; Zbl 0593.51016), p. 355]. The results are: (1) In some pseudoaffine space \((P,{\mathcal L})\) there exists a line which is not central, although its set of parallels is left fixed by some translation (:=automorphism having no fixed points) of \((P,{\mathcal L})\). (\(L\in {\mathcal L}\) is called central iff \(X\| L\), \(Y\| L\) implies \(X\| Y\) for all \(X,Y\in {\mathcal L}\).) (2) Any pseudoaffine Euclidean space must be affine. (3) Let (H,\(\cdot)\) be a group of exponent 3. \(x,y:=\{x,y,x\bar y^ 1x\}\), \({\mathcal L}:=\{\overline{x,y}: x,y\in H\), \(x\neq y\}\). Subspaces of the pseudoaffine space \((H,{\mathcal L})\) are called z-pseudoaffine, and the author characterizes them by the following Theorem: Let \((P,{\mathcal L})\) be a partial line space. Equivalent are: (a) \((P,{\mathcal L})\) is \(z\)-pseudoaffine. (b) \((P,{\mathcal L})\) is coordinatized by some commutative Moufang loop of exponent 3 and nilpotency class at most 2. (c) For any \(A,C\in {\mathcal L}\), \(A\| C\), and \(x\in P\) such that \(\{x\| A\}\neq \{x\| C\}\) (\(\{p\| L\}\) denotes the unique line parallel to \(L\), containing \(p\)), the plane generated by \(\{x\| A\|\}\), \(\{x\| C\}\) contains (up to parallelism) exactly one central line.
reviewed by: Horst Szambien
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 4042684
zbMATH Keywords: pseudoaffine spaces
zbMATH Keywords: Moufang loops
zbMATH Keywords: z-pseudoaffine
author: Kay Soerensen
MaRDI profile type: Publication
cites work: Une caractérisation des espaces affins basée sur la notion de droite
cites work: Q3253828
cites work: Incidence axioms for affine geometry
cites work: Translationsstrukturen, die weder axial noch zentral sind
cites work: Endliche Gruppen I
cites work: Q3723210
cites work: Bericht über geschlitzte Inzidenzgruppen
cites work: Q5662517
cites work: Q4078247
cites work: Pseudo-Euklidische Ebenen und Euklidische Räume
cites work: Inzidenzgruppen der Ordnung 3
full work available at URL: https://doi.org/10.1007/bf01222397
OpenAlex ID: W2911286294
Recommended article: Q3684735
Recommended article: Q4944315
Recommended article: Q3706084
Recommended article: Representation of hyperplanes in generalized affine spaces by modules
Recommended article: Planar and affine spaces
Recommended article: Q4950112
Recommended article: Q4546169
Recommended article: A note on affine subspaces
Recommended article: An axiom system for partial affine spaces
Recommended article: Q3324542
description: scientific article; zbMATH DE number 4042684