A proof of the bundle theorem for certain semimodular locally projective lattices of rank 4
type: Item
instance of: scholarly article
title: A proof of the bundle theorem for certain semimodular locally projective lattices of rank 4
zbMATH Open document ID: 0575.06009
DOI: 10.1016/0097-3165(85)90038-X
author: Rolfdieter Frank
published in: Journal of Combinatorial Theory. Series A
publication date: 1985-01-01T00:00:00Z
review text: Let L be a projective geometry of rank n and let Q be a nonempty set of points of L. Denote \(L(Q)=\{x\in L:\) \(\exists p\in Q\) with \(p\leq x\}\cup \{0\}\). \textit{R. Wille} [Atti Conv. Geom. Combinat. Appl. Perugia 1970, 421-426 (1971; Zbl 0256.50003)] has shown that every semimodular locally projective lattice of rank \(n>4\), such that the whole space is the join of the set of its points, is projectively embeddedable, i.e. is isomorphic to some L(Q). But the conclusion is false for lattices of rank 4. In the paper, the author obtains an affirmative result for geometric locally projective lattices of rank 4 under some additional assumption which requires, roughly speaking, that for each line \(\ell\) there are not too many lines which are on a common plane with \(\ell\), but on no common point.
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 51A05
zbMATH DE Number: 3918430
zbMATH Keywords: projective geometry
zbMATH Keywords: semimodular locally projective lattice
zbMATH Keywords: projectively embeddedable
zbMATH Keywords: geometric locally projective lattices
MaRDI profile type: Publication
cites work: Q5331549
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description: scientific article; zbMATH DE number 3918430