Projective remoteness planes
type: Item
instance of: scholarly article
title: Projective remoteness planes
zbMATH Open document ID: 0848.51004
DOI: 10.1007/BF00147363
author: John R. Faulkner
published in: Geometriae Dedicata
publication date: 1996-10-28T00:00:00Z
review text: The author gives a set of axioms involving incidence and remoteness (the negation of neighboring) leading to a class of Barbilian planes called projective remoteness planes. It is shown that a transvection projective remoteness plane \(P\) can be coordinatized by a ring \(R\) which is a two-sided inverse ring. Moreover it is proved that \(P\) is a transvection plane iff \(P\) is isomorphic to \(P(G,N)\), the plane associated with a group \(G\) of Steinberg type parametrized by \(R\) and with \(N\) a certain subgroup of \(G\). The author investigates also necessary and sufficient conditions on \(P\) for \(R\) to be alternative, associative or commutative. The notion of elementary \(n\)-basis set is introduced and constructions of projective and affine remoteness planes are given. Finally a plane with reflections determining a system of rotations is shown to have commutative, associative coordinates.
Mathematics Subject Classification ID: 51C05
zbMATH DE Number: 874400
zbMATH Keywords: coordinatization
zbMATH Keywords: Steinberg group
zbMATH Keywords: Barbilian planes
zbMATH Keywords: projective remoteness planes
reviewed by: Dirk Keppens
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147363
OpenAlex ID: W1966219538
description: scientific article; zbMATH DE number 874400