Principal congruences on some lattice-ordered algebras
type: Item
instance of: scholarly article
title: Principal congruences on some lattice-ordered algebras
zbMATH Open document ID: 0702.06011
DOI: 10.1016/0012-365X(90)90068-S
author: J. C. Varlet
author: T. S. Blyth
published in: Discrete Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The authors solve the question in which algebras all congruences are principal (PC). They prove: 1) A distributive lattice L has property (PC) if L is finite and all subsets of J(L) \((=\) the poset of all nonzero join-irreducible elements of L) are convex (resp. \(\ell (J(L))\leq 1).\) 2) A Stone algebra L has property (PC) if L is finite and \(\ell (J(L))\leq 2.\) 3) A de Morgan algebra has property (PC) if L is finite and \(\ell (J(L))\leq 3.\) They also show that the Heyting algebras behave quite differently - to ensure that all congruences are principal a chain condition is necessary and sufficient.
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 06D30
zbMATH DE Number: 4150243
zbMATH Keywords: principal congruences
zbMATH Keywords: distributive lattice
zbMATH Keywords: join-irreducible elements
zbMATH Keywords: Stone algebra
zbMATH Keywords: de Morgan algebra
zbMATH Keywords: Heyting algebras
zbMATH Keywords: chain condition
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4150243