On a-semiaffine planes with invisible lines
type: Item
instance of: scholarly article
title: On a-semiaffine planes with invisible lines
zbMATH Open document ID: 0651.51011
DOI: 10.1007/BF00181604
author: Albrecht Beutelspacher
author: Inge Schestag
published in: Geometriae Dedicata
publication date: 1987-01-01T00:00:00Z
review text: The aim of the paper is to study finite weakly a-semiaffine planes. Suppose that the line set of a linear space L is partitioned into two sets, the `visible' and the `invisible' lines. For a non-incident point- line pair (P,\(\ell)\) let \(\pi\) (P,\(\ell)\) denote the number of lines through P which do not intersect \(\ell\). A weakly a-semiaffine plane is a linear space L where (among others) the following conditions hold: For \(P\not\in \ell\) we have \(\pi (P,\ell)\geq a-\ell;\) moreover \(\pi\) (P,\(\ell)\leq a\) iff \(\ell\) is visible. The main results give statements concerning the degrees of points and visible lines. For example, if L is of order n then every point has degree \(n+1\) or n and every visible line has degree \(n+2-a\) or \(n+1-a\). Remark: If the set of invisible lines is empty then L is an a-semiaffine plane [see the first author: Einführung in die endliche Geometrie. II: Projektive Räume. (1983; Zbl 0502.51001)].
Mathematics Subject Classification ID: 51E30
zbMATH DE Number: 4062032
zbMATH Keywords: finite linear spaces
zbMATH Keywords: finite weakly a-semiaffine planes
reviewed by: Wolfgang Nolte
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181604
OpenAlex ID: W2092258312
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description: scientific article; zbMATH DE number 4062032