The determination of subvarieties of certain congruence-distributive varieties
type: Item
instance of: scholarly article
title: The determination of subvarieties of certain congruence-distributive varieties
zbMATH Open document ID: 0816.08008
DOI: 10.1007/BF01190816
published in: Algebra Universalis
publication date: 1995-07-20T00:00:00Z
review text: The author develops a project concerned with the computer generation of equational bases for varieties of distributive-lattice-ordered algebras. Let \(\mathcal K\) be such a variety generated by a finite algebra. The general problem is to determine a set of identities which, added to a fixed identity basis of \(\mathcal K\), suffices to define via conjunctions every nontrivial subvariety of \(\mathcal K\). It naturally divides into three subproblems: (1) Find the lattice of all subvarieties of \(\mathcal K\) (it is known to be isomorphic to the lattice of order-ideals of the set of isomorphism types of subdirectly irreducible algebras of \(\mathcal K\) ordered by the relation \(A\leq B\to A\in {\mathbf H}{\mathbf S} B)\). (2) Find a procedure which, when applied to any specified covering pair \(V',V''\subseteq {\mathcal K}\), outputs an identity distinguishing \(V'\) from \(V''\). (3) Find a procedure for converting the local information supplied by (2) into global information. The paper under review presents a strategy for (3) based on the well- known distributive lattice duality due to the author [Bull. Lond. Math. Soc. 2, 186-190 (1970; Zbl 0201.018), Proc. Lond. Math. Soc., III. Ser. 24, 507-530 (1972; Zbl 0323.06011)]. The algorithm is illustrated by finding the desired identity set for the variety of Ockham algebras with de Morgan skeleton.
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 06D30
Mathematics Subject Classification ID: 08-04
Mathematics Subject Classification ID: 06-04
zbMATH DE Number: 679609
zbMATH Keywords: congruence distributivity
zbMATH Keywords: lattice of subvarieties
zbMATH Keywords: equational bases for varieties of distributive-lattice-ordered algebras
zbMATH Keywords: duality
zbMATH Keywords: algorithm
zbMATH Keywords: Ockham algebras
zbMATH Keywords: de Morgan skeleton
author: Hilary A. Priestley
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01190816
OpenAlex ID: W2093425591
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description: scientific article; zbMATH DE number 679609