Relation algebras as residuated Boolean algebras
type: Item
instance of: scholarly article
title: Relation algebras as residuated Boolean algebras
zbMATH Open document ID: 0792.06012
DOI: 10.1007/BF01195378
author: Bjarni Jónsson
author: Constantine Tsinakis
published in: Algebra Universalis
publication date: 1994-08-08T00:00:00Z
review text: After some general remarks on residuated and conjugate operations in a Boolean algebra, a residuated Boolean algebra is defined as a Boolean algebra that is augmented with three binary operations, \(\circ\), \(\vartriangleleft\), and \(\vartriangleright\), such that for all elements \(x\), \(y\), and \(z\), \((x\circ y)\cdot z= 0\) iff \((x\vartriangleright z)\cdot 0\) iff \((z\vartriangleleft y)\cdot x= 0\). Such an algebra is unital if there is an identity element \(e\) for \(\circ\). Every nonassociative relation algebra gives rise to a unital residuated Boolean algebra by defining \(x\circ y= x;y\), \(x\vartriangleleft y= x;\breve y\) and \(x\vartriangleleft y= \breve x;y\). Several characterizations are given for those unital residuated Boolean algebras arising in this way. Several equations equivalent to the associativity of \(\circ\) are given, along with characterizations of those unital residuated Boolean algebras that come from relation algebras.
Mathematics Subject Classification ID: 06E25
Mathematics Subject Classification ID: 03G15
zbMATH DE Number: 488263
zbMATH Keywords: residuals
zbMATH Keywords: conjugates
zbMATH Keywords: residuated Boolean algebra
zbMATH Keywords: nonassociative relation algebra
reviewed by: Roger Maddux
MaRDI profile type: Publication
Wikidata QID: Q29396197
cites work: Q5532570
cites work: Q3763567
cites work: Some Varieties Containing Relation Algebras
cites work: The Concept of Join System and Its Basic Properties
full work available at URL: https://doi.org/10.1007/bf01195378
OpenAlex ID: W2065137642
description: scientific article; zbMATH DE number 488263