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The Categories of Boolean Lattices, Boolean Rings and Boolean Spaces

Published online by Cambridge University Press:  20 November 2018

Hoshang P. Doctor
Hoshang P. Doctor*
Affiliation:
McMaster University
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In Theorem 4 of [5] Stone stated that the theory of Boolean rings was "mathematically equivalent" to the theory of Boolean spaces without, however, properly defining the phrase "mathematically equivalent". It is the main purpose of this note to establish a precise reformulation of Theorem 4 in [5]. This is accomplished by introducing special classes of maps between Boolean lattices, Boolean rings and Boolean spaces respectively, and showing the categories arising in conjunction with these maps to be equivalent in the sense of Grothendieck [2]. Thus the notion of equivalence of categories will replace the phrase "mathematically equivalent" in [5]. In addition the well-known axiomatic characterization of meet and complementation of Boolean lattices with unit is discussed in analogous terms.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 2 , June 1964 , pp. 245 - 252
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Eilenberg, S. and MacLane, S., General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945), 231–294.Google Scholar
2. Grothendieck, A., Sur quelques points d'algebre homologique, Tôhoku Math. J. 9 (1957), 119–121.Google Scholar
3. Rosenbloom, P., The elements of mathematical logic, New York, Dover Publications, Inc., 1950.Google Scholar
4. Stone, M. H., Theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 37–111.Google Scholar
5. Stone, M. H., Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375–481.Google Scholar