The lattice of subsemilattices of a semilattice
type: Item
instance of: scholarly article
title: The lattice of subsemilattices of a semilattice
zbMATH Open document ID: 0797.06003
DOI: 10.1007/BF01236520
author: Ilya B. Muchnik
author: Leonid Libkin
published in: Algebra Universalis
publication date: 1994-04-12T00:00:00Z
review text: The main result of this paper is the following characterization of the subsemilattice-lattice: A lattice \(\mathcal L\) is isomorphic to \(\text{Sub}(L)\) for some semilattice \(L\) iff it is algebraic, biatomic (that is, for any two non-zero \(x,y\in {\mathcal L}\) and any atom \(z\leq x\lor y\) there exist atoms \(x'\leq x\), \(y'\leq y\) such that \(z\leq x'\lor y'\)) and satisfies \((S_ 3)\) (we say that a biatomic lattice \(\mathcal L\) satisfies properties \((S_ n)\) if for any ideal \(V\) generated by \(n\) atoms \(a_ 1,\dots,a_ n\in {\mathcal L}\) there exists a finite semilattice \(L_ V\) such that \(V\simeq\text{Sub}(L_ V)\) and the natural embedding of ideals \(V\mapsto W\) induces the embedding of semilattices \(L_ V\mapsto L_ W\).
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 549249
zbMATH Keywords: algebraic lattice
zbMATH Keywords: biatomic lattice
zbMATH Keywords: ideals
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
cites work: Biatomic lattices
cites work: Q3875978
description: scientific article; zbMATH DE number 549249