Congruence lattices of \(p\)-algebras
type: Item
instance of: scholarly article
title: Congruence lattices of \(p\)-algebras
zbMATH Open document ID: 0827.06007
DOI: 10.1007/BF01225470
published in: Algebra Universalis
publication date: 1995-07-31T00:00:00Z
review text: A \(p\)-algebra is a bounded lattice endowed with a unary operation \({}^*\) satisfying \(x\leq a^*\) iff \(x\wedge a=0\). The authors give a new proof of the following theorem due to Katriňák: Every finite distributive lattice is isomorphic to the congruence lattice of a finite \(p\)-algebra. Then they indicate how to modify the proof so as to obtain the following generalization. If \(D\) is an algebraic distributive lattice such that the unit element of \(D\) is compact and every compact element of \(D\) is a finite join of join-irreducible compact elements, then \(D\) can be represented as the congruence lattice of a \(p\)-algebra. Finally some open problems are stated.
Mathematics Subject Classification ID: 06D15
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 780793
zbMATH Keywords: finite distributive lattice
zbMATH Keywords: congruence lattice
zbMATH Keywords: \(p\)-algebra
zbMATH Keywords: algebraic distributive lattice
zbMATH Keywords: compact element
author: G. Grätzer
author: E. T. Schmidt
reviewed by: Sergiu Rudeanu
MaRDI profile type: Publication
cites work: Q3875978
cites work: Congruence lattices of planar lattices
cites work: On congruence lattices of lattices
cites work: Every finite distributive lattice is the congruence lattice of some modular lattice
cites work: Congruence lattices of finite \(p\)-algebras
description: scientific article; zbMATH DE number 780793