Weakly independent subsets in lattices
type: Item
instance of: scholarly article
title: Weakly independent subsets in lattices
zbMATH Open document ID: 0569.06006
DOI: 10.1007/BF01278596
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: A subset H of a lattice L is weakly independent if for all \(h,h_ 1,h_ 2,...,h_ n\in H\) which satisfy \(h\leq h_ 1\vee h_ 2\vee...\vee h_ n\) there must exist an index i (1\(\leq i\leq n)\) such that \(h\leq h_ i\). A basis is a maximal weakly independent set. Let \(J_ 0(L)\) denote the set of join irreducible elements of the lattice L. The following are the main results of the paper. Theorem 1. Any two bases of a finite distributive lattice have the same cardinality. Theorem 2. A subset H of a finite distributive lattice L is a basis for L if and only if the sublattice K generated by H is a sublattice of maximal length and \(H=J_ 0(K)\). The paper closes with an example of a modular lattice for which both Theorems fail.
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B05
zbMATH DE Number: 3908492
zbMATH Keywords: basis
zbMATH Keywords: maximal weakly independent set
zbMATH Keywords: join irreducible elements
zbMATH Keywords: finite distributive lattice
zbMATH Keywords: sublattice of maximal length
author: Gábor Czédli
author: Andras P. Huhn
author: E. T. Schmidt
reviewed by: Melvin F. Janowitz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf01278596
OpenAlex ID: W2027543491
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description: scientific article; zbMATH DE number 3908492