Axiomatization of identity-free equations valid in relation algebras
type: Item
instance of: scholarly article
title: Axiomatization of identity-free equations valid in relation algebras
zbMATH Open document ID: 0852.03024
DOI: 10.1007/BF01195500
published in: Algebra Universalis
publication date: 1996-06-02T00:00:00Z
review text: Two new finite axiom systems are given for the identity-free equations valid in a relation algebra. They consist of the identity-free axioms of a relation algebra (i.e., the axioms distinct \(x\circ 1'= 1'\circ x=x\)) and one of the axioms \(x\leq x\circ ((y^\cup \circ y^-)^- \cdot (y^{-\cup} \circ y)^-)\), \(x\leq x\circ ((y^\cup \circ y^-)^- \cdot (z^\cup \circ z^- )^-)\); the latter axioms cannot be replaced by \(x\leq x\circ (y^\cup \circ y^-)\). This answers problems raised by Tarski and Jónsson. Further, let \(\text{RRA}^{\circ,\cup}\) be the class of all algebras of binary relations, up to isomorphism, whose operations are the Boolean ones, relation composition and the operation of taking converses, and let \(\text{RRA}^\circ\) be defined similarly but without the operation \({}^\cup\). Then \(\text{RRA}^{\circ, \cup}\) is finitely axiomatizable over \(\text{RRA}^\circ\).
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 08A02
zbMATH DE Number: 883421
zbMATH Keywords: finite axiom systems
zbMATH Keywords: identity-free equations
zbMATH Keywords: relation algebra
zbMATH Keywords: algebras of binary relations
author: Hajnal Andréka
author: István Németi
reviewed by: Sergiu Rudeanu
MaRDI profile type: Publication
cites work: Perfect extensions and derived algebras
cites work: Varieties of relation algebras
cites work: Adjoining units to residuated Boolean algebras
cites work: Boolean Algebras with Operators. Part I
cites work: Q3708022
cites work: A sequent calculus for relation algebras
full work available at URL: https://doi.org/10.1007/bf01195500
OpenAlex ID: W1487689712
Recommended article: Q4000536
Recommended article: Equational axiomatization of algebras with structure
Recommended article: The equational theory of union-free algebras of relations
Recommended article: Complexity of equations valid in algebras of relations. II: Finite axiomatizations
Recommended article: Axiomatizability of reducts of algebras of relations
Recommended article: Q4273431
Recommended article: Q4348001
Recommended article: Q4943493
Recommended article: Notes on equational theories of relations
Recommended article: Q3222193
description: scientific article; zbMATH DE number 883421