Direct decompositions of atomistic algebraic lattices
type: Item
instance of: scholarly article
title: Direct decompositions of atomistic algebraic lattices
zbMATH Open document ID: 0818.06004
DOI: 10.1007/BF01190769
published in: Algebra Universalis
publication date: 1995-02-22T00:00:00Z
review text: A lattice is atomistic if each of its elements is a join of atoms. An element \(x\) of a complete lattice \(L\) is strictly join-irreducible if \(x= \bigvee X\) implies \(x\in X\) for any subset \(X\subseteq L\). A lattice \(L\) is called a \(V_ 1\)-lattice if each its elements is the join of strictly join-irreducible elements of \(L\). Main results: Theorem 1. Every atomistic algebraic lattice is a direct product of directly indecomposable (atomistic algebraic) lattices. Theorem 2. Every algebraic \(V_ 1\)-lattice is a direct product of directly indecomposable (algebraic) \(V_ 1\)-lattices.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 724106
zbMATH Keywords: atomistic lattice
zbMATH Keywords: directly indecomposable lattice
zbMATH Keywords: strictly join- irreducible elements
zbMATH Keywords: algebraic lattice
author: Leonid Libkin
MaRDI profile type: Publication
cites work: Convexity lattices
cites work: Q4165427
cites work: Direct product decompositions of lattices, closures and relation schemes
cites work: The lattice of subsemilattices of a semilattice
cites work: On the structure of lattices in which every element is a join of join-irreducible elements
description: scientific article; zbMATH DE number 724106