Lattice geometries
type: Item
instance of: scholarly article
title: Lattice geometries
zbMATH Open document ID: 0795.06012
DOI: 10.1007/BF01195376
published in: Algebra Universalis
publication date: 1994-09-12T00:00:00Z
review text: The Frink embedding theorem [\textit{O. Frink jun.}, Trans. Am. Math. Soc. 60, 452-467 (1946; Zbl 0060.058)] by which complemented modular lattices are embedded into a union of subspace lattices of projective planes and suitable irreducible projective coordinate spaces is generalized by Theorem 2 of \textit{U. Faigle} [Arch. Math. 36, 179-182 (1981; Zbl 0452.06005)] which says that any modular lattice can be embedded in a complete modular point lattice whose points (= join-irreducible elements) are compact and which satisfies all the identities of the embedded lattice. The authors give a generalization of Faigle's result: They define a \(J\)- additive lattice geometry to be a pair \((L,J)\), where \(L\) is a lattice and \(J\subset L\) such that any element in \(L\) is the join of all elements in \(J\) beneath it, and any \((x_ 1,x_ 2)\in L\times L\) is \(J\)-additive which means that whenever \(p\in J\), \(p\leq x_ 1\lor x_ 2\), there exist \(q_ 1,q_ 2\in J\), \(q_ i\leq x_ i\), \(i\in \{1,2\}\) such that \(p\leq q_ 1\lor q_ 2\). As examples of \(J\)-additive lattice geometries there are given among others (1) the lattice of submodules of any unital left module with \(J\) to be taken as the set of cyclic submodules, and (2) the biatomic lattices defined by the second author [Algebra Univers. 24, 60-73 (1987; Zbl 0643.06003)], where \(J\) is the set of atoms. The lattice of closed subspaces of an infinite-dimensional Hilbert space with \(J\) chosen to be the one-dimensional subspaces is not \(J\)-additive. Their main result is then: For any \(J\)-additive lattice geometry \((L,J)\) the underlying lattice \(L\) can be embedded under preservation of lattice identities into its lattice \(L'\) of flats which is then complete and algebraic. Furthermore \((L',J')\) is \(J'\)-additive, where \(J'\) denotes the image of \(J\) under the embedding. Here a flat \(F\) is defined to be subset of \(J\) such that \(x,y\in F\), \(z\in J\), \(z\leq x\lor y\) always implies \(z\in F\)…
Mathematics Subject Classification ID: 06C99
Mathematics Subject Classification ID: 51D25
Mathematics Subject Classification ID: 06C05
zbMATH DE Number: 488261
zbMATH Keywords: projective geometry
zbMATH Keywords: modular lattice
zbMATH Keywords: \(J\)-additive lattice geometry
author: R. Piziak
author: Mary Katherine Bennett
reviewed by: Horst Szambien
MaRDI profile type: Publication
cites work: Biatomic lattices
cites work: Projective Geometry on Partially Ordered Sets
cites work: Frink's theorem for modular lattices
cites work: Modulare Verbände mit Punktsystem
cites work: Complemented Modular Lattices and Projective Spaces of Infinite Dimension
cites work: Q4165427
cites work: On the modular relation in atomistic lattices
cites work: Q5623714
cites work: Spaces of type n on partially ordered sets
cites work: Projective spaces on partially ordered sets and Desargues' postulate
cites work: Halbkomplementäre Verbände
full work available at URL: https://doi.org/10.1007/bf01195376
OpenAlex ID: W4248382553
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description: scientific article; zbMATH DE number 488261