On interpretations of varieties with semilattice reduct
type: Item
instance of: scholarly article
title: On interpretations of varieties with semilattice reduct
zbMATH Open document ID: 0728.08010
DOI: 10.1007/BF01182463
published in: Algebra Universalis
publication date: 1990-01-01T00:00:00Z
review text: A variety V is interpretable in a variety W (V\(\leq W)\) if for every fundamental V-operation \(f_ t\), \(t\in T\), there exists a W-term \(\alpha_ t\) such that for every algebra \(A\in W\) the algebra \((A;\alpha_ t,t\in T)\in V\). The assignment I: \(f_ t\mapsto \alpha_ t\), \(t\in T\), is called an interpretation of V in W. The paper delivers a simplified approach of results of R. Levin and of W. Taylor on interpretations of distributive lattices in Heyting algebras and of n- element linearly ordered Heyting algebra in \(n+1\)-element linearly ordered Heyting algebras. Among other results the following is shown for varieties V, W of Heyting algebras which are generated by three finite members. If \(V\leq W\) and W properly contains the variety of Boolean algebras then there is exactly one interpretation of V in W. If \(V\nleq W\) then V is not interpretable in W.
Mathematics Subject Classification ID: 08B99
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 4202345
zbMATH Keywords: interpretability of a variety in another one
zbMATH Keywords: Heyting algebras
author: W. J. Blok
MaRDI profile type: Publication
cites work: Brouwerian Semilattices
cites work: Interpretations into Heyting algebras
cites work: A note on interpretations of Heyting algebras
description: scientific article; zbMATH DE number 4202345