On sublattice lattice varieties
type: Item
instance of: scholarly article
title: On sublattice lattice varieties
zbMATH Open document ID: 0723.06008
DOI: 10.1007/BF01190720
author: Dang X. Hong
published in: Algebra Universalis
publication date: 1990-01-01T00:00:00Z
review text: In this note the author gives solutions to Problem I.6 and Problem I.7 in \textit{G. Grätzer}'s book: General lattice theory (1978; Zbl 0436.06001). The solutions are as follows: (1) For any non-trivial lattice variety \({\mathcal K}\), Sub(\({\mathcal K})\) is the variety of all lattices. (2) The only lattice varieties \({\mathcal K}\) for which Sub(\({\mathcal K})={\mathcal K}\) are the variety of all lattices and the trivial variety of the one- element lattice. Here Sub(\({\mathcal K})\) denotes the variety generated by the class of all sublattice lattices of members of \({\mathcal K}\). The key to the problems is the following theorem: Any finite lattice can be embedded into the sublattice of a finite Boolean lattice.
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 4191756
zbMATH Keywords: lattice variety
zbMATH Keywords: sublattice lattices
zbMATH Keywords: Boolean lattice
reviewed by: Yuichi Komori
MaRDI profile type: Publication
cites work: Q3875978
cites work: Every finite lattice can be embedded in a finite partition lattice
cites work: Lattices, equivalence relations, and subgroups
full work available at URL: https://doi.org/10.1007/bf01190720
OpenAlex ID: W2094420733
description: scientific article; zbMATH DE number 4191756