Do we need complete-simple distributive lattices?
type: Item
instance of: scholarly article
title: Do we need complete-simple distributive lattices?
zbMATH Open document ID: 0819.06008
DOI: 10.1007/BF01190771
published in: Algebra Universalis
publication date: 1995-02-22T00:00:00Z
review text: A complete lattice is said to be complete-simple if it has no nontrivial complete congruence relations. Let \(L\) be a complete lattice; if \(\theta\) is a complete congruence of \(L\), then \(L/\theta\) is called a complete quotient of \(L\). The authors prove the following result: Let \(C(D)\) be the lattice of all complete congruences of a complete distributive lattice \(D\). Suppose that either (i) \(C(D)\) is a three-element chain, or (ii) \(C(D)\) is the five-element nonmodular lattice. Then \(D\) has an infinite distributive complete-simple lattice as a complete quotient.
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06B10
zbMATH DE Number: 724108
zbMATH Keywords: complete lattice
zbMATH Keywords: complete congruence
zbMATH Keywords: complete quotient
zbMATH Keywords: distributive lattice
zbMATH Keywords: complete-simple lattice
author: G. Grätzer
author: E. T. Schmidt
reviewed by: Ján Jakubík
MaRDI profile type: Publication
cites work: Q4165427
cites work: Q4286322
full work available at URL: https://doi.org/10.1007/bf01190771
OpenAlex ID: W1980373098
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description: scientific article; zbMATH DE number 724108