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Groups and Algebras of Binary Relations

Published online by Cambridge University Press:  15 January 2014

Steven Givant  and
Hajnal Andréka
Steven Givant
Affiliation:
Department of Mathematics and Computer Science, Mills College, 5000 Macarthur Boulevard, Oakland, CA 94613, USAE-mail: givant@mills.edu
Hajnal Andréka
Affiliation:
Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest P. O. Box 127, H-1364 Hungary, E-mail: andreka@renyi.hu
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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 Jónsson 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 Jónsson 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 Jónsson 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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 8 , Issue 1 , March 2002 , pp. 38 - 64
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1] Birkhoff, G., On the combination of subalgebras, Proceedings of the Cambridge Philosophical Society, vol. 29 (1933), pp. 441–464.Google Scholar
[2] Boole, G., The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning, MacMillan, Barclay, and MacMillan, London, 1847, 82 pp.Google Scholar
[3] Chin, L. H. and Tarski, A., Distributive and modular laws in the arithmetic of relation algebras, University ofCalifornia Publications in Mathematics, New Series, vol. 1 (1951), pp. 341–384.Google Scholar
[4] de Morgan, A., On the syllogism, no. IV, and on the logic of relations, Transactions of the Cambridge Philosophical Society, vol. 10 (1864), pp. 331–358.Google Scholar
[5] Jónsson, B. and Tarski, A., Representation problems for relation algebras, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 80 and 1192, Abstract 89.Google Scholar
[6] Jónsson, B. and Tarski, A., Boolean algebras with operators. Part I, American Journal of Mathematics, vol. 73 (1951), pp. 891–939.Google Scholar
[7] Jónsson, B. and Tarski, A., Boolean algebras with operators. Part II, American Journal of Mathematics, vol. 74 (1952), pp. 127–162.Google Scholar
[8] Löwenheim, L., Einkleidung der Mathematik in Schroderschen Relativkalkul, The Journal of Symbolic Logic, vol. 5 (1940), pp. 1–15.Google Scholar
[9] Lyndon, R. C., The representation of relational algebras, Annals of Mathematics, vol. 51 (1950), pp. 707–729.CrossRefGoogle Scholar
[10] Maddux, R., Pair-dense relation algebras, Transactions of the American Mathematical Society, vol. 328 (1991), pp. 83–131.CrossRefGoogle Scholar
[11] Monk, J. D., On representable relation algebras, Michigan Mathematical Journal, vol. 11 (1964), pp. 207–210.CrossRefGoogle Scholar
[12] Monk, J. D., Completions of Boolean algebras with operators, Mathematische Nachrichten, vol. 46 (1970), pp. 47–55.CrossRefGoogle Scholar
[13] Peirce, C. S., Note B. The logic of relatives, Studies in logic by members of the Johns Hopkins University (Peirce, C. S., editor), Little, Brown, and Company, Boston, 1883, [Reprinted by John Benjamins Publishing Company, Amsterdam, 1983], pp. 187–203.CrossRefGoogle Scholar
[14] Schröder, E., Vorlesungen über die Algebra der Logik (exakte Logik), vol. III, Algebra und Logik der Relative , part 1, B. G. Teubner, Leipzig, 1895, 649 pp. [Reprinted by Chelsea Publishing Company, New York, 1966.]Google Scholar
[15] Tarski, A., On the calculus of relations, The Journal of Symbolic Logic, vol. 6 (1941), pp. 73–89.CrossRefGoogle Scholar
[16] Tarski, A., Contributions to the theory of models. I, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, vol. 57 (1954) (= Indagationes Mathematicae , vol. 16), pp. 572–581.Google Scholar
[17] Tarski, A., Contributions to the theory of models. II, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, vol. 57 (1954) (= Indagationes Mathematicae , vol. 16), pp. 582–588.Google Scholar
[18] Tarski, A., Contributions to the theory of models. III, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, vol. 58 (1955) (= Indagationes Mathematicae , vol. 17), pp. 56–64.Google Scholar
[19] Tarski, A. and Givant, S., A formalization of set theory without variables, Colloquium Publications, vol. 41, American Mathematical Society, Providence, RI, 1987, xxii + 318 pp.Google Scholar