Non-modular varieties of semimodular lattices with a spanning \(M_ 3\)
type: Item
instance of: scholarly article
title: Non-modular varieties of semimodular lattices with a spanning \(M_ 3\)
zbMATH Open document ID: 0558.06005
DOI: 10.1016/0012-365X(85)90142-6
author: Robert W. Quackenbush
published in: Discrete Mathematics
publication date: 1985-01-01T00:00:00Z
review text: There are several lattice constructions which go back to the tensor product \(A\otimes B\) in the category of join semilattices with zero (the morphisms are zero-preserving join homomorphisms). The author considers the tensor product of modular lattices to obtain special semimodular lattices. L is a lattice with spanning \(M_ 3\) if L contains 5 nullary operations a,b,c,0,1 so that \(a\wedge b=a\wedge c=b\wedge c=0\) and \(a\vee b=a\vee c=b\vee c=1\). The main results are: 1. For any bounded modular lattice B, \(M_ 3\otimes B\) is isomorphic to the lattice of all 1- preserving meet homomorphisms from \(M_ 3\) into B, and it is a semimodular lattice with a spanning \(M_ 3\). 2. Let \({\mathcal K}\) be a variety of modular lattices, \({\mathcal K}_ 3\) is the class of all lattices L with 5 nullary operations isomorphic to \(M_ 3\otimes B\) for some B in \({\mathcal K}\). Then \({\mathcal K}_ 3\) is a variety of semimodular lattices with spanning \(M_ 3\). 3. As category \({\mathcal K}_ 3\) is equivalent to \({\mathcal K}\), moreover \({\mathcal K}_ 3\) is finitely based iff \({\mathcal K}\) is finitely based. 4. If A is a bounded semimodular lattice and D is a bounded distributive lattice then \(A\otimes D\) is semimodular. 5. \(M_ 4\otimes M_ 4\) is not semimodular. 6. \(M_ 3\otimes M_ 3\) is not supersolvable (a lattice is supersolvable if it contains a maximal chain of modular elements). Some open problems are listed.
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 06C10
zbMATH DE Number: 3889593
zbMATH Keywords: tensor product
zbMATH Keywords: category of join semilattices
zbMATH Keywords: modular lattices
zbMATH Keywords: semimodular lattices
zbMATH Keywords: bounded modular lattice
zbMATH Keywords: variety of modular lattices
reviewed by: Tamás Schmidt
MaRDI profile type: Publication
cites work: The tensor product of semilattices
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cites work: Supersolvable lattices
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full work available at URL: 0012-365x(85)90142-6
OpenAlex ID: W2068423935
description: scientific article; zbMATH DE number 3889593