Biatomic lattices
type: Item
instance of: scholarly article
title: Biatomic lattices
zbMATH Open document ID: 0643.06003
DOI: 10.1007/BF01188384
published in: Algebra Universalis
publication date: 1987-01-01T00:00:00Z
review text: The author studies ``biatomic'' lattices which are atomic lattices L satisfying: \(p\in L\) atom, a, \(b\in L\), \(p\leq a\vee b\) imply \(p\leq c\vee d\) for atoms \(c\leq a\) and \(d\leq b\). The subspace lattice of a vector space is biatomic. For an ordered division ring D the lattice of convex subsets of D n is biatomic. Let C \(n\subseteq D\) n be the n-cube. Then its face lattice \({\mathcal F}(C\) n) is biatomic. Products and ideals of biatomic lattices are biatomic. The MacNeille completion of a biatomic lattice is biatomic. For a biatomic lattice L there exists in the variety generated by L a complete algebraic biatomic lattice which contains L as a sublattice. For an atomic lattice L are equivalent: (i) L is modular, (ii) L is biatomic and has the exchange property.
Mathematics Subject Classification ID: 06B05
zbMATH DE Number: 4047785
zbMATH Keywords: atomic lattices
zbMATH Keywords: lattice of convex subsets
zbMATH Keywords: face lattice
zbMATH Keywords: biatomic lattices
author: Mary Katherine Bennett
reviewed by: Gudrun Kalmbach
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01188384
OpenAlex ID: W4236366453
description: scientific article; zbMATH DE number 4047785