The embedding of \((0,\alpha)\)-geometries in \(\mathrm{PG}(n,q)\). II
type: Item
instance of: scholarly article
title: The embedding of \((0,\alpha)\)-geometries in \(\mathrm{PG}(n,q)\). II
zbMATH Open document ID: 0548.51006
DOI: 10.1016/0012-365X(84)90009-8
published in: Discrete Mathematics
publication date: 1984-01-01T00:00:00Z
review text: For a positive integer \(\alpha >1,\) a finite (0,\(\alpha)\)-geometry is a connected incidence structure \(S=(P,B,I)\) with a symmetric incidence relation satisfying the fllowing axioms: (i) two distinct points are incident with at most one line, (ii) if a point x and a line L are not incident, then there are 0 or \(\alpha\) points which are collinear with x and incident with L, and (iii) each line is incident with at least two points, and each point is incident with at least two lines. By the connectedness of S one can prove that there exists a positive integer \(s>1\) such that every point is incident with \(s+1\) lines and, duality, that each line is incident with \(s+1\) points. In the paper under review the authors continue the study of (0,\(\alpha)\)- geometries previously initiated in [\textit{J. A. Thas} and \textit{F. De Clerck}, Ann. Discrete Math. 18, 229-240 (1983; Zbl 0503.51004)]. First, they determine all (0, \(\alpha)\)-geometries with \(q+1\) points on a line embeddable in PG(n,q), where \(n>3\) and \(q>2\) is assumed. Second, they prove some theorems about the embedding of (0,2)-geometries in PG(n,2). In particular, they determine all (0,2)-geometries in PG(4,2) and describe an unusual embedding of \(U_{2,3}(9)\) in PG(5,2).
reviewed by: J.Libicher
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 3874085
zbMATH Keywords: (0,alpha)-geometries
author: Joseph A. Thas
author: Ingrid Debroey
author: F. De Clerck
MaRDI profile type: Publication
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full work available at URL: 0012-365x(84)90009-8
OpenAlex ID: W2044060188
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description: scientific article; zbMATH DE number 3874085