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ON TARSKI’S AXIOMATIC FOUNDATIONS OF THE CALCULUS OF RELATIONS

Published online by Cambridge University Press:  08 May 2017

HAJNAL ANDRÉKA ,
STEVEN GIVANT ,
PETER JIPSEN  and
ISTVÁN NÉMETI
HAJNAL ANDRÉKA
Affiliation:
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS HUNGARIAN ACADEMY OF SCIENCES H-1364BUDAPEST, PF. 127 HUNGARYE-mail: andreka.hajnal@renyi.mta.hu
STEVEN GIVANT
Affiliation:
MILLS COLLEGE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE 5000 MACARTHUR BOULEVARD OAKLAND, CA 94613, USAE-mail:givant@mills.edu
PETER JIPSEN
Affiliation:
CHAPMAN UNIVERSITY, FACULTY OF MATHEMATICS SCHOOL OF COMPUTATIONAL SCIENCES 545 WEST PALM AVENUE ORANGE, CA 92866, USAE-mail:jipsen@chapman.edu
ISTVÁN NÉMETI
Affiliation:
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS HUNGARIAN ACADEMY OF SCIENCESH-1364BUDAPEST, PF. 127 HUNGARYE-mail:nemeti.istvan@renyi.mta.hu
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Abstract

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system.

Keywords

03G1503B3003C0503C13axiom systemcalculus of relationsindependence modelindependent axiom systemrelation algebra
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 966 - 994
Copyright
Copyright © The Association for Symbolic Logic 2017 

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