Representable biresiduated lattices
type: Item
instance of: scholarly article
title: Representable biresiduated lattices
zbMATH Open document ID: 1001.06012
published in: Journal of Algebra
publication date: 2002-12-15T00:00:00Z
review text: A biresiduated lattice is an algebra \({\mathbf A}=\langle A;\cdot ,/,\backslash ,\wedge ,\vee ,1\rangle \) such that \(\langle A;\wedge,\vee \rangle \) is a lattice in which \(1\) is the maximum element, and \( /\) and \(\backslash \) are binary operations which satisfy the following ``residuation'' properties with respect to the lattice order: \(a\cdot c\leq b\) iff \(c\leq a\backslash b\); \(c\cdot a\leq b\) iff \(c\leq b/a\) for all \(a\), \(b\), \(c\) in \(A\). The corresponding class is denoted by \({\mathcal B}\). These algebras were first studied by Krull as abstractions of ideal lattices of rings enriched with the monoid operation of ideal multiplication. The objective of this paper is to axiomatize the class of all the biresiduated lattices that may be represented as subalgebras of products of linearly ordered biresiduated lattices (these are called representable and the corresponding class is denoted by \({\mathcal L}\)). It is shown that \({\mathcal L}\) is axiomatized, relative to \({\mathcal B}\), by the identity \[ (x\backslash y)\vee ([w\cdot (z\backslash ((y\backslash x)\cdot z))]/w)=1 \] or, equivalently, by \[ (x\backslash y)\vee (w/(w/(((y\backslash x)\backslash z)\backslash z)))=1. \] Representable cancellative and complemented biresiduated lattices are also discussed. Finally, connections with lattice-ordered groups are given.
Mathematics Subject Classification ID: 06F10
Mathematics Subject Classification ID: 06F15
zbMATH DE Number: 1740520
zbMATH Keywords: axiomatization
zbMATH Keywords: biresiduated lattice
author: C. J. van Alten
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
full work available at URL: jabr.2001.9039
OpenAlex ID: W2168541099
cites work: Q4039742
cites work: Lattice-ordered groups
cites work: Varieties of commutative residuated integral pomonoids and their residuation subreducts
cites work: THE STRUCTURE OF RESIDUATED LATTICES
cites work: Q3934450
cites work: Subdirect products of totally ordered BCK-algebras
cites work: Über halbgeordnete Gruppen
cites work: Logics without the contraction rule
cites work: Q3874204
cites work: Q3932964
cites work: Q3775561
cites work: Q4943158
DOI: 10.1006/JABR.2001.9039
description: scientific article; zbMATH DE number 1740520