On hyperplanes and free subspaces of affine Klingenberg spaces
type: Item
instance of: scholarly article
title: On hyperplanes and free subspaces of affine Klingenberg spaces
zbMATH Open document ID: 0817.51005
DOI: 10.1007/BF01832980
published in: Aequationes Mathematicae
publication date: 1994-12-07T00:00:00Z
full work available at URL: https://eudml.org/doc/137602
review text: Previously [in Combinatorics, Ravello 1988, 185-200, Rende (1991)], the authors described two equivalent axiom systems for affine Klingenberg spaces \({\mathcal A} = (P, {\mathcal L}, \|, \sim)\). In the present paper, they study one of these systems in more detail. A maximal independent subset of \({\mathcal A}\) maps to a maximal independent subset of the ordinary affine space \({\mathcal A}^* = {\mathcal A}/ \sim\). A subspace \({\mathcal S}\) of \({\mathcal A}\) is said to be free if \({\mathcal S}\) is spanned by an independent set of points. It follows that \(\dim {\mathcal S} = \dim\;{\mathcal S}^*\). A subspace \({\mathcal H}\) is called a hyperplane if \({\mathcal H} = \langle M \backslash \{h\} \rangle\) for some independent point set \(M\) and \(h \in M\). Equivalently, \({\mathcal H}\) is free and \({\mathcal H}^*\) is a hyperplane of \({\mathcal A}^*\). In order to obtain basic properties of hyperplanes , two additional axioms are introduced; one requires that a hyperplane and a non-neighbouring line meet in at most 1 point. Freeness of \({\mathcal A}\) can then be characterized in terms of geometric properties of hyperplanes, but there remain several open questions.
Mathematics Subject Classification ID: 51D15
zbMATH DE Number: 699772
zbMATH Keywords: affine Klingenberg spaces
zbMATH Keywords: hyperplanes
author: T. Bisztriczky
author: J. W. Michael Lorimer
reviewed by: Helmut Salzmann
MaRDI profile type: Publication
cites work: Q4150278
cites work: Q4505257
cites work: Hjelmslev-Räume. (Hjelmslev spaces)
cites work: Free modules over Hjelmslev rings in which not every maximal linearly independent subset is a basis
cites work: Projektive Hjelmslevräume.
cites work: Affine Klingenbergsche Strukturen
cites work: Q3906703
Recommended article: Q4026370
Recommended article: Subspace operations in affine Klingenberg spaces
Recommended article: On the \(K\)-theory of affine hyperplanes through a linear subvariety of codimension one
Recommended article: A Khintchine-type theorem for affine subspaces
Recommended article: Q5479998
Recommended article: Affine spaces and algebras of subalgebras
Recommended article: On k-Subspaces of L-vector-spaces
Recommended article: Affine Laumon spaces and a conjecture of Kuznetsov
Recommended article: Q3723201
Recommended article: A note on affine subspaces
description: scientific article; zbMATH DE number 699772