# Relation algebras and groups

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## Abstract

Generalizing results of Jónsson and Tarski, Maddux introduced the notion of a *pair-dense* relation algebra and proved that every pair-dense relation algebra is representable. The notion of a pair below the identity element is readily definable within the equational framework of relation algebras. The notion of a triple, a quadruple, or more generally, an element of size (or measure) \(n>2\) is not definable within this framework, and therefore it seems at first glance that Maddux’s theorem cannot be generalized. It turns out, however, that a very far-reaching generalization of Maddux’s result is possible if one is willing to go outside of the equational framework of relation algebras, and work instead within the framework of the first-order theory. Moreover, this generalization sheds a great deal of light not only on Maddux’s theorem, but on the earlier results of Jónsson and Tarski. In the present paper, we define the notion of an atom below the identity element in a relation algebra having measure *n* for an arbitrary cardinal number \(n>0\), and we define a relation algebra to be *measurable* if it’s identity element is the sum of atoms each of which has some (finite or infinite) measure. The main purpose of the present paper is to construct a large class of new examples of *group relation algebras* using systems of groups and corresponding systems of quotient isomorphisms (instead of the classic example of using a single group and forming its complex algebra), and to prove that each of these algebras is an example of a measurable set relation algebra. In a subsequent paper, the class of examples will be greatly expanded by adding a third ingredient to the mix, namely systems of “shifting” cosets. The expanded class of examples—called *coset relation algebras*—will be large enough to prove a representation theorem saying that every atomic, measurable relation algebra is essentially isomorphic to a coset relation algebra.

## Keywords

Relation algebra Group Representable relation algebra Measurable relation algebra Group relation algebra## Mathematics Subject Classification

03G15 03E20 20A15## Notes

### Acknowledgements

The author is very much indebted to Dr. Hajnal Andréka, of the Alfréd Rényi Mathematical Institute in Budapest, for carefully reading a draft of this paper and making many extremely helpful suggestions.

## References

- 1.Andréka, H., Givant, S.: Coset relation algebras. Algebra Universalis
**(in press)**Google Scholar - 2.De Morgan, A.: On the syllogism, no. IV, and on the logic of relations. Trans. Camb. Philos. Soc.
**10**, 331–358 (1864)Google Scholar - 3.Givant, S.: Introduction to Relation Algebras. Springer International Publishing AG, Cham (2017)CrossRefMATHGoogle Scholar
- 4.Givant, S., Andréka, H.: Groups and algebras of relations. Bull. Symb. Logic
**8**, 38–64 (2002)MathSciNetCrossRefMATHGoogle Scholar - 5.Givant, S., Andréka, H.: A representation theorem for measurable relation algebras
**(to appear)**Google Scholar - 6.Givant, S., Andréka, H.: Simple Relation Algebras. Springer International Publishing AG, Cham (2017)CrossRefMATHGoogle Scholar
- 7.Hirsch, R., Hodkinson, I.: Relation algebras by games. Studies in Logic and the Foundations of Mathematics, vol. 147. Elsevier Science, North-Holland Publishing Company, Amsterdam (2002)Google Scholar
- 8.Jónsson, B., Tarski, A.: Representation problems for relation algebras. Bull. Am. Math. Soc. Abstr.
**54**, 801192 (1948)Google Scholar - 9.Jónsson, B., Tarski, A.: Boolean algebras with operators. Part II. Am. J. Math.
**74**, 127–162 (1952)CrossRefMATHGoogle Scholar - 10.Lyndon, R.C.: The representation of relational algebras. Ann. Math.
**51**, 707–729 (1950)MathSciNetCrossRefMATHGoogle Scholar - 11.Maddux, R.D.: Pair-dense relation algebras. Trans. Am. Math. Soc.
**328**, 83–131 (1991)MathSciNetCrossRefMATHGoogle Scholar - 12.Maddux, R.D.: Relation algebras. Studies in Logic and the Foundations of Mathematics, vol. 150. Elsevier Science, North-Holland Publishing Company, Amsterdam (2006)Google Scholar
- 13.Monk, J.D.: On representable relation algebras. Mich. Math. J.
**11**, 207–210 (1964)MathSciNetCrossRefMATHGoogle Scholar - 14.Peirce, C.S.: Note B. The logic of relatives. In: Peirce, C.S. (ed.) Studies in logic by members of the Johns Hopkins University, pp. 187–203. Little, Brown, and Company, Boston (1883) [Reprinted by John Benjamins Publishing Company, Amsterdam (1983)]Google Scholar
- 15.Schröder, E.: Vorlesungen über die Algebra der Logik (exakte Logik), vol. III. Algebra und Logik der Relative, part 1. B.G. Teubner, Leipzig (1895) [Reprinted by Chelsea Publishing Company, New York (1966)]Google Scholar
- 16.Tarski, A.: On the calculus of relations. J. Symb. Logic
**6**, 73–89 (1941)MathSciNetCrossRefMATHGoogle Scholar - 17.Tarski, A.: Contributions to the theory of models. III. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, Mathematical Sciences
**58**[Indagationes Mathematicae**17**, 56–64 (1955)]Google Scholar