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Projective algebra and the calculus of relations1

Published online by Cambridge University Press:  12 March 2014

A. R. Bednarek  and
S. M. Ulam
A. R. Bednarek
Affiliation:
University of Florida, Gainesville, Florida 32611
S. M. Ulam
Affiliation:
University of Florida, Gainesville, Florida 32611
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Extract

In [1] there were given postulates for an abstract “projective algebra” which, in the words of the authors, represented a “modest beginning for a study of logic with quantifiers from a boolean point of view”. In [5], D. Monk observed that the study initiated in [1] was an initial step in the development of algebraic versions of logic from which have evolved the cylindric and polyadic algebras.

Several years prior to the publication of [1], J. C. C. McKinsey [3] presented a set of postulates for the calculus of relations. Following the publication of [1], McKinsey [4] showed that every projective algebra is isomorphic to a subalgebra of a complete atomic projective algebra and thus, in view of the representation given in [1], every projective algebra is isomorphic to a projective algebra of subsets of a direct product, that is, to an algebra of relations.

Of course there has since followed an extensive development of projective algebra resulting in the multidimensional cylindric algebras [2]. However, what appears to have been overlooked is the correspondence between the Everett–Ulam axiomatization and that of McKinsey.

It is the purpose of this paper to demonstrate the above, that is, we show that given a calculus of relations as defined by McKinsey it is possible to introduce projections and a partial product so that this algebra is a projective algebra and conversely, for a certain class of projective algebras it is possible to define a multiplication so that the resulting algebra is McKinsey's calculus of relations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 1 , March 1978 , pp. 56 - 64
Copyright
Copyright © Association for Symbolic Logic 1978

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Footnotes

1

Research supported by NSF Grant No. MSC75–21130.

References

REFERENCES

[1]Everett, C. J. and Ulam, S. M., Projective algebra. I, American Journal of Mathematics, vol. 68 (1946), pp. 77–88.CrossRefGoogle Scholar
[2]Henkin, L., Monk, D., and Tarski, A., Cylindric algebras, Part I, North-Holland, Amsterdam, 1971.Google Scholar
[3]McKinsey, J. C. C., Postulates for the calculus of binary relations, this Journal, vol. 5 (1940), pp. 85–97.Google Scholar
[4]McKinsey, J. C. C., On the representation of projective algebras, American Journal of Mathematics, vol. 70 (1948), pp. 375–384.CrossRefGoogle Scholar
[5]Stanislaw Ulam: Sets, Numbers, and Universes (Beyer, W. A., Mycielski, J., and Rota, G. C., Editors), MIT Press, 1974.Google Scholar
[6]Tarski, A., On the calculus of relations, this Journal, vol. 6 (1941), pp. 73–89.Google Scholar