A geometric description of modular lattices
type: Item
instance of: scholarly article
title: A geometric description of modular lattices
zbMATH Open document ID: 0816.06008
DOI: 10.1007/BF01221792
author: Christian Herrmann
author: Douglas Pickering
author: Micheale Susan Roddy
published in: Algebra Universalis
publication date: 1995-07-23T00:00:00Z
review text: The authors consider modular lattices \(M\) which are spatial, i.e. every element is a join of some subset of the set \(S(M)\) of points (= completely join irreducible elements) of \(M\). If \(S(M)\) is given the induced partial order \(\leq\) and a collinearity relation \(C\) where \(C(p, q,r)\) is defined by \(p+ q= p+r= q+r\) for any pairwise incomparable \(p,q,r\in S(M)\) then \(S(M)\) constitutes a so-called ordered space in which subspaces are then defined as those order ideals which are closed with respect to collineraity. The resulting lattice of subspaces \(L(S(M))\) contains \(M\), and it is even isomorphic to \(M\) if \(M\) is algebraic (2.4). Taking up investigations from \textit{U. Faigle} and the first author [Trans. Am. Math. Soc. 266, 319-332 (1981; Zbl 0466.51001)] an ordered space \((S,\leq C)\) is called projective if it satisfies a line regularity axiom and a kind of a Veblen-Young axiom which includes some degenerate situations. Such ordered spaces are shown to arise exactly from spatial modular lattices (3.2, 3.3). Furthermore, any variety of modular lattices is generated by subspace lattices of countable ordered projective spaces (6.2). The work is done in continuation of the first author and \textit{M. Wild} [J. Algebra 136, 17-36 (1991; Zbl 0725.16014)] by carefully selecting necessary conditions (chain conditions, countability) on the point set and so-called line bases which allow to reconstruct the lattice structure in the case of modular algebraic spatial lattices. Finally, the authors point out properties which are equivalent to 2- distributivity in modular spatial lattices \(M\); one of these properties is the absence of the triangle configurations occurring in the Veblen- Young axiom (7.2), see also \textit{B. Jónsson} and \textit{J. B. Nation} [Acta Sci. Math. 51, 123-128 (1987; Zbl 0633.06002)]. They show that as in the case of finite length, treated by B. Jonsson and J. B. Nation [loc. cit], 2-distributive lattices can be embedded i…
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 51A05
zbMATH DE Number: 590886
zbMATH Keywords: order ideals
zbMATH Keywords: spatial modular lattices
zbMATH Keywords: subspace lattices
zbMATH Keywords: ordered projective spaces
zbMATH Keywords: 2-distributive lattices
reviewed by: Horst Szambien
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590886