On the construction of finite projective planes from homology semibiplanes
type: Item
instance of: scholarly article
title: On the construction of finite projective planes from homology semibiplanes
zbMATH Open document ID: 0729.51008
DOI: 10.1016/S0195-6698(13)80044-3
published in: European Journal of Combinatorics
publication date: 1990-01-01T00:00:00Z
review text: From a projective plane \(\Pi\) with involutory homology \(\tau\) one constructs an incidence system \(\Pi/\tau\) having as points and blocks the \(<\tau>\)-orbits of length 2 on the points and lines of \(\Pi\), and with incidence inherited from \(\Pi\). Such incidence systems satisfy certain properties which, when taken as axioms, define the class of homology semibiplanes. We describe how one determines, in principle, whether a given homology semibiplane \(\Sigma\) is realizable as \(\Pi/\tau\) for some \(\Pi\) and \(\tau\) and, moreover, how many non-equivalent pairs \((\Pi,\tau)\) yield \(\Sigma\). In case \(\Pi '\) is Desarguesian of prime order we show that \(\Pi'\) is characterized by its homology semibiplane; i.e. \(\Pi/\tau \simeq \Pi'/\tau'\) implies \(\Pi \simeq \Pi'\). The details are too involved to be described here. For the definition of a semibiplane we follow here that employed by \textit{D. R. Hughes} and \textit{F. C. Piper} [``Design Theory'' (1985; Zbl 0561.05009), pp. 196-211].
reviewed by: Giorgio Faina
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 4205265
zbMATH Keywords: projective plane
zbMATH Keywords: semibiplane
author: G. Eric Moorhouse
MaRDI profile type: Publication
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full work available at URL: s0195-6698(13)80044-3
OpenAlex ID: W2097559003
description: scientific article; zbMATH DE number 4205265