On maximal sublattices of finite lattices
type: Item
instance of: scholarly article
title: On maximal sublattices of finite lattices
zbMATH Open document ID: 0941.06013
DOI: 10.1016/S0012-365X(98)00207-6
published in: Discrete Mathematics
publication date: 2000-07-27T00:00:00Z
review text: The author investigates the relationships between finite distributive lattices, their maximal sublattices and the corresponding remainders. If \(L\) is a lattice and \(M\) a maximal sublattice of \(L\), then \(L\) is a minimal extension of \(M\), and the remainder \(R\) is defined as \(R= L\setminus M\). \(R\) is a sublattice of \(L\) whenever \(L\) is distributive. It is shown that a finite distributive lattice \(M\) has only finitely many minimal extensions \(L\) preserving 0 and 1. An algorithm is given to effectively list the minimal extensions. In contrast, given a finite distributive lattice \(R\), there are infinitely many pairs \((M,L)\) so that \(R\approx L\setminus M\). Again, a list of these pairs can be generated effectively. Birkhoff duality is the tool used to establish these results. It is known that \(|R|\leq|L|/3\). In this paper it is shown that equality occurs exactly if \(L\approx K\times 3\) and \(M\approx K\times 2\) for some finite distributive lattice \(K\). The author cites \textit{M. E. Adams, P. Dwinger} and \textit{J. Schmid}, ``Maximal sublattices of finite distributive lattices'' [Algebra Univers. 36, No. 4, 488-504 (1996; Zbl 0901.06009)] and \textit{M. E. Adams, R. Freese, J. B. Nation} and \textit{J. Schmid}, ``Maximal sublattices and Frattini sublattices of bounded lattices'' [J. Austral. Math. Soc., Ser. A 63, No. 1, 110-127 (1997; Zbl 0887.06005)] and the references given in these two papers as a port of entry to the literature on the subject. For general finite lattices, not necessarily distributive, questions about extensions and remainders have been addressed in the second of these papers. Some of the results of that paper are discussed at the end of the present paper and some open questions are discussed.
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 1321798
zbMATH Keywords: finite distributive lattices
zbMATH Keywords: maximal sublattices
zbMATH Keywords: remainders
zbMATH Keywords: minimal extensions
zbMATH Keywords: algorithm
zbMATH Keywords: Birkhoff duality
author: Jürg Schmid
MaRDI profile type: Publication
cites work: The Frattini sublattice of a distributive lattice
cites work: Maximal sublattices of finite distributive lattices
cites work: Q4361780
cites work: Finite distributive lattices and doubly irreducible elements
cites work: Frattini sublattices of distributive lattices
cites work: Q4692885
cites work: Q5814847
cites work: Maximal Sublattices of Finite Distributive Lattices
cites work: Boolean layer cakes
cites work: Q4004146
full work available at URL: s0012-365x(98)00207-6
OpenAlex ID: W2087406706
Recommended article: Q5501721
Recommended article: Minimal representations of a finite distributive lattice by principal congruences of a lattice
Recommended article: The structure of finite distributive lattices.
Recommended article: Almost distributive lattice varieties
Recommended article: Q5302119
Recommended article: Q5701825
Recommended article: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices
Recommended article: Q3433170
Recommended article: Q3695311
Recommended article: Modularity and distributivity of 3-generated lattices with special elements among generators
description: scientific article; zbMATH DE number 1321798