Planar and affine spaces
type: Item
instance of: scholarly article
title: Planar and affine spaces
zbMATH Open document ID: 1193.51014
author: Pınar Anapa
author: İbrahim Günaltili
author: Hendrik Van Maldeghem
published in: Discrete Mathematics
publication date: 2009-12-09T00:00:00Z
review text: Three-dimensional affine spaces are characterized as the only linear spaces endowed with a set \(\Omega\) of proper subspaces having the following properties: (1) Every line contains a constant number \(n\), \(n>2\), of points. (2) Every triple of noncollinear points is contained in a unique member of \(\Omega\). (3) `Disjoint or coincide' is an equivalence relation in \(\Omega\), and every equivalence class covers all points. The case \(n=2\) (when the structure is a complete graph endowed with a set \(\Omega\) of proper complete subgraphs) is also considered. It turns out that besides the affine 3-space of order 2, only two small additional examples occur. Finally, the authors characterize all finite affine spaces of dimension at least 3 having lines of size at least 3.
reviewed by: József Szilasi
Mathematics Subject Classification ID: 51E26
zbMATH DE Number: 5644079
zbMATH Keywords: planar spaces
zbMATH Keywords: affine spaces
zbMATH Keywords: linear spaces
MaRDI profile type: Publication
full work available at URL: j.disc.2008.07.024
OpenAlex ID: W2026213351
cites work: Q3719375
cites work: Projective planar spaces
cites work: Une caractérisation des espaces affins basée sur la notion de droite
cites work: Q4764138
cites work: Finite regular locally projective spaces
cites work: On regular planar spaces of type (\(k,n\))
cites work: A rank inequality for finite geometric lattices
cites work: Combinatorial properties of planar spaces and embeddability
DOI: 10.1016/J.DISC.2008.07.024
description: scientific article; zbMATH DE number 5644079