Varieties of commutative residuated integral pomonoids and their residuation subreducts
type: Item
instance of: scholarly article
title: Varieties of commutative residuated integral pomonoids and their residuation subreducts
zbMATH Open document ID: 0872.06007
author: W. J. Blok
author: J. G. Raftery
published in: Journal of Algebra
publication date: 1997-09-21T00:00:00Z
review text: Let \(\langle A;\oplus ,0,\leq \rangle\) be a commutative (dually) integral partially ordered monoid whose identity \(0\) is the least element of \(\langle A,\leq \rangle\), where \(\leq\) is a partial order compatible with the monoid operation \(\oplus\) in the sense that \(a\oplus b\leq c\oplus d\) whenever \(a\leq c\) and \(b\leq d\). If for each \(a,b\in A\) there is a least element \(c\in A\) (denoted \(a\div b\)) such that \(a\leq c\oplus b\), then the resulting structure \(\langle A;\oplus ,\div ,0,\leq \rangle\) is called a partially ordered commutative (dually) residuated (dually) integral monoid (briefly a pocrim). The operation \( \div \) is called residuation. BCK-algebras are the residuation subreducts of pocrims. The simplest example of a pocrim is the set of ideals of a commutative ring with \(1\) with respect to the ideal multiplication as the monoid operation and the (lattice) order of reversed set inclusion. Residuation is defined by \( I\div J=I:J\). The class of all pocrims is a quasivariety \( \mathcal{V} \) which is not a variety, but it is relatively congruence distributive, has the relative congruence extension property, and is relatively point regular with respect to the monoid identity \(0\). All \(0\)-regular subvarieties are congruence \(n\)-permutable for some \(n\). In several varieties of pocrims (for example in the variety of Brouwerian semilattices and the variety of hoops) the finitely generated subdirectly irreducible algebras can be obtained from a finite number of simple algebras in the variety by a finite number of applications of the operations of variety generation and ordinal sum. Cancellative pocrims form a subquasivariety \( \mathcal{C} \) which has a Mal'cev term and generates a congruence permutable variety (not contained in \( \mathcal{V} \)). The subvarieties of \( \mathcal{C} \) form a lattice which has no greatest element.
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 06F35
zbMATH DE Number: 1014000
zbMATH Keywords: BCK-algebra
zbMATH Keywords: residuation
zbMATH Keywords: variety
zbMATH Keywords: quasivariety
zbMATH Keywords: partially ordered monoid
zbMATH Keywords: subreducts
zbMATH Keywords: ideals of a commutative ring
zbMATH Keywords: relatively congruence distributive
zbMATH Keywords: relative congruence extension property
zbMATH Keywords: Brouwerian semilattices
zbMATH Keywords: hoops
zbMATH Keywords: subdirectly irreducible algebras
zbMATH Keywords: congruence permutable
reviewed by: Wiesław A. Dudek
MaRDI profile type: Publication
full work available at URL: jabr.1996.6834
OpenAlex ID: W2083174445
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Varieties of commutative residuated integral pomonoids and their residuation subreducts (DOI)
DOI: 10.1006/JABR.1996.6834