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The Categories of Boolean Lattices, Boolean Rings and BooleanSpaces

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 themain purpose of this note to establish a precise reformulation of Theorem 4in [5]. This is accomplished by introducing special classes of maps betweenBoolean lattices, Boolean rings and Boolean spaces respectively, and showingthe categories arising in conjunction with these maps to be equivalent inthe sense of Grothendieck [2]. Thus the notion of equivalence of categorieswill replace the phrase "mathematically equivalent" in [5]. In addition thewell-known axiomatic characterization of meet and complementation of Booleanlattices with unit is discussed in analogous terms.

Information

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
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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