Duality in stable planes and related closure and kernel operations
type: Item
instance of: scholarly article
title: Duality in stable planes and related closure and kernel operations
zbMATH Open document ID: 0926.51015
DOI: 10.1007/BF01229209
author: Holger Bickel
published in: Journal of Geometry
publication date: 1999-11-21T00:00:00Z
review text: Stable planes are topological incidence geometries with a ``local'' planarity condition (stability) replacing the requirement that any two lines meet (or the parallel axiom). Examples include all geometries induced on open nonempty sets of points in topological projective planes, in particular, the hyperbolic planes. As usual, the paper under review concentrates on the locally compact case. The involution assigning to each projective plane its dual is extended, in two different ways, to subclasses of the class of stable planes. If \(E\) is an open subset of the point set \(P\) of a topological projective plane \((P,{\mathcal L})\), one obtains a stable plane \(E^{\text{dual}}=({\mathcal E},P)\) as subplane of \(({\mathcal L},P)\). The largest subset of \(P\) whose dual coincides with \(E^{\text{dual}}\) is denoted \(E^*\). The paper under review shows \(E^{**}=E^*\). For an arbitrary stable plane \((E,{\mathcal E})\) (without embedding into a projective plane) one considers the set \({\mathcal K}\) of all projective lines (that is, lines that meet all other lines). The author defines \(E^\circ\) as the set of all points lying on some projective line, and shows \(E^{\circ\circ}=E^\circ\). It is known that \(E^\circ\) is an open subset of~\(E\), so \(E^\circ\) is the point space of a stable plane \(E^{\text{opp}}:=({\mathcal K},E^\circ)\) if \(\mathcal K\) is not empty. In the paper under review, it is shown that \(\Aut(E^{\text{dual}})=\Aut(E^*)\) and that \(\Aut(E^{\text{opp}})=\Aut(E^\circ)\) if \(\mathcal K\) is not empty. These results are used to determine the automorphism groups of complex cylinder planes.
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 1292214
zbMATH Keywords: stable plane
zbMATH Keywords: opposite plane
zbMATH Keywords: dual plane
zbMATH Keywords: automorphism group
reviewed by: Markus Stroppel
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01229209
OpenAlex ID: W2056915876
description: scientific article; zbMATH DE number 1292214