Varieties generated by lattices of breadth two
type: Item
instance of: scholarly article
title: Varieties generated by lattices of breadth two
zbMATH Open document ID: 0804.06010
DOI: 10.1007/BF01111297
author: Jörg Stephan
published in: Order
publication date: 1995-01-16T00:00:00Z
review text: A lattice \(L\) is called dismantlable if every non-empty sublattice of \(L\) contains a doubly irreducible (i.e. join-irreducible and meet- irreducible) element. It is shown that a dismantlable lattice is a lattice of breadth \(\leq 2\). Some relations between the class \({\mathcal B}\) of all lattices of breadth \(\leq 2\) and the class \({\mathcal D}\) of all dismantlable lattices, as well as the lattice varieties \({\mathbf V} ({\mathcal B})\) and \({\mathbf V} ({\mathcal D})\) generated by \({\mathcal B}\) and \({\mathcal D}\), resp., are studied in the paper. It is noted that \({\mathbf V} ({\mathcal B})\) satisfies \((x\vee y)\wedge (x\vee z)= x\vee ((x\vee y)\wedge (x\vee z)\wedge (y\vee z))\). The main results are as follows. A lattice of breadth \(\leq n\) is \(n\)-distributive (Proposition 2). For finite join- semidistributive (meet-semidistributive) lattice concepts of dismantlability and breadth \(\leq 2\) coincide (Proposition 1). On the other hand, the interval \([{\mathbf V} ({\mathcal D}), {\mathbf V} ({\mathcal B})]\) of the lattice of all lattice varieties contains continuum many varieties (Theorem 1), and all varieties of this interval are not finitely based (Theorem 2).
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 540214
zbMATH Keywords: dismantlable lattice
zbMATH Keywords: breadth
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: Equational classes of modular lattices
cites work: Equational axioms for classes of lattices
cites work: Q3937460
cites work: Q3956998
cites work: Drawing orders with few slopes
cites work: Weak (projective) radius and finite equational bases for classes of lattices
cites work: Q5686073
cites work: Q3872518
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: On the importance of the relation \([(A,B),(A,C)]=(A,[(B,C),(C,A),(A,B)])\) between three elements of a structure
cites work: Equational Bases and Nonmodular Lattice Varieties
cites work: Lattices with Doubly Irreducible Elements
cites work: Combinatorial inequalities for semimodular lattices of breadth two
full work available at URL: https://doi.org/10.1007/bf01111297
OpenAlex ID: W2126461494
description: scientific article; zbMATH DE number 540214