On algebras and varieties with semilattice reducts
type: Item
instance of: scholarly article
title: On algebras and varieties with semilattice reducts
zbMATH Open document ID: 0787.08009
DOI: 10.1007/BF01196100
author: Margarita Ramalho
published in: Algebra Universalis
publication date: 1994-05-17T00:00:00Z
review text: \textit{W. Blok} [ibid. 27, 299-303 (1990; Zbl 0728.08010)] proved how some results of \textit{R. Lewin} [ibid. 24, 149-166 (1987; Zbl 0614.06009)], on interpretations in the variety of Heyting algebras, could be derived from a lemma concerning finite algebras having a special property with respect to a reduct in the variety \({\mathcal {SL}}_ 1\) of all semilattices with neutral element. The first aim of this note is to extend this lemma to all such algebras -- called by Blok semilattices with filter-preserving operations. As a consequence, we obtain that the lattice of subvarieties of Heyting algebras is dually embeddable in the lattice of interpretability types of varieties. In Algebra, logic and number theory, Proc. 15th Port.-Span. Meet. Math., Evora/Port. 1990, Vol. I, 37-42 (1991; Zbl 0743.06004), the author gave a characterization of algebras with a reduct in \({\mathcal {SL}}\) having a property stronger than the one considered by Blok; namely, the property \((H)\) of having a congruence lattice isomorphic to the lattice of all filters of a semilattice reduct, under an isomorphism which maps each congruence \(\theta\) to a \(\theta\)- class. We now characterize those varieties \({\mathcal V}\) having the \(H\)- property with respect to an interpretation of \({\mathcal {SL}}_ 1\) in \({\mathcal V}\).
Mathematics Subject Classification ID: 08B99
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 08A99
Mathematics Subject Classification ID: 06A12
zbMATH DE Number: 203157
zbMATH Keywords: interpretations in the variety of Heyting algebras
zbMATH Keywords: semilattices with filter-preserving operations
zbMATH Keywords: lattice of subvarieties of Heyting algebras
zbMATH Keywords: lattice of interpretability types of varieties
zbMATH Keywords: congruence lattice
zbMATH Keywords: semilattice reduct
MaRDI profile type: Publication
cites work: On interpretations of varieties with semilattice reduct
cites work: Algebraizable logics
cites work: The lattice of interpretability types of varieties
cites work: Brouwerian Semilattices
cites work: Interpretations into Heyting algebras
cites work: Q4011728
cites work: Completeness in arithmetical algebras
description: scientific article; zbMATH DE number 203157