Maximal and completely meet irreducible sublattices of a distributive lattice
type: Item
instance of: scholarly article
title: Maximal and completely meet irreducible sublattices of a distributive lattice
zbMATH Open document ID: 0537.06011
DOI: 10.1007/BF01194539
author: Raymond Balbes
published in: Algebra Universalis
publication date: 1983-01-01T00:00:00Z
review text: For a lattice L we use the following notations: \({\mathcal P}(L)\) resp. \({\mathcal Q}(L)\) for the set of all prime ideals resp. filters of L and (x] resp. [x) the ideal resp. filter generated by x. A lattice is distributive iff it contains sufficiently many prime ideals (enough to separate every two elemnts); Stone (1936); Iseki (1951). Using prime filters too and extending a work of Hashimoto (1952) the author proves a separation property for a pair: element-sublattice: Lemma 2.1. - Let S be a sublattice of a distributive lattice. For \(x\in L\backslash S\), each of the following holds: (i) If \(S\cap [x)=\emptyset\) then there exists \(I\in {\mathcal P}(L)\) such that \(x\not\in I\) and \(S\subseteq I\); (ii) If \(S\cap(x]=\emptyset\) then there exists \(F\in {\mathcal Q}(L)\) such that \(x\not\in F\) and \(S\subseteq F\); (iii) If \(S\cap [x)\neq \emptyset\) and \(S\cap(x]\neq \emptyset\) then there exist \(I\in {\mathcal P}(L)\) and \(F\in {\mathcal Q}(L)\) such that \(x\not\in I\cup F\) and \(S\subseteq I\cup F\). - From this lemma the main result follows easily. Theorem 2.2. - Let L be a distributive lattice. A subset \(\emptyset \neq S\subseteq L\) is a proper sublattice of L iff there exist index sets \(J_ 1,J_ 2,J_ 3\) such that \[ S=(\cap_{j\in J_ 1}I_ j)\cap(\cap_{j\in J_ 2}F_ j)\cap(\cap_{j\in J_ 3}(I_ j\cup F_ j)) \] where \(I_ j\in {\mathcal P}(L)\), \(F_ j\in {\mathcal Q}(L)\), \(I_ j\cup F_ j\neq L\) for all \(j\in J_ 3\) and \(J_ 1\cup J_ 2\cup J_ 3\neq \emptyset.\) This characterization is very prolific. The author proves characterizations for maximal and, more generally, for completely meet irreducible sublattices and applies them for finite distributive lattices, Boolean algebras, free distributive lattices and Post algebras of order \(n\geq 2\), recovering results of Rival, Koh, Chen and Tan concerning maximal sublattices. We mention Theorem 4.2. - Let S be a sublattice of a distributive lattice L. Then S is completely meet irreducible iff o…
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B10
zbMATH DE Number: 3853159
zbMATH Keywords: prime ideals
zbMATH Keywords: filters
zbMATH Keywords: completely meet irreducible sublattices
zbMATH Keywords: finite distributive lattices
zbMATH Keywords: Boolean algebras
zbMATH Keywords: free distributive lattices
zbMATH Keywords: Post algebras
MaRDI profile type: Publication
cites work: Q4137227
cites work: Frattini sublattices of distributive lattices
cites work: Q5814847
cites work: On the Frattini sublattice of a lattice
cites work: Relation of the Exact Transient Solution for a Line Source near an Interface between Two Fluids to Geometrical Acoustics
cites work: Maximal Sublattices of Finite Distributive Lattices
cites work: Maximal Sublattices of Finite Distributive Lattices. II
description: scientific article; zbMATH DE number 3853159