Bulletin of Symbolic Logic

Groups and Algebras of Binary Relations

Steven Givant and Hajnal Andreka

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Abstract

In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras, He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jonsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jonsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms-called coset relation algebras-that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well, We prove that every atomic relation algebra satisfying a certain measurability condition-a condition generalizing the conditions imposed by Jonsson and Tarski-is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations.

Article information

Source
Bull. Symbolic Logic, Volume 8, Number 1 (2002), 38-64.

Dates
First available in Project Euclid: 20 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.bsl/1182353852

Digital Object Identifier
doi:10.2178/bsl/1182353852

Mathematical Reviews number (MathSciNet)
MR1888166

Zentralblatt MATH identifier
1002.03055

JSTOR
links.jstor.org

Citation

Givant, Steven; Andreka, Hajnal. Groups and Algebras of Binary Relations. Bull. Symbolic Logic 8 (2002), no. 1, 38--64. doi:10.2178/bsl/1182353852. https://projecteuclid.org/euclid.bsl/1182353852


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