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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Homomorphic images of $\sigma$-complete Boolean algebras
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by Sabine Koppelberg PDF
Proc. Amer. Math. Soc. 51 (1975), 171-175 Request permission

Abstract:

It is a well-known theorem of R. S. Pierce that, for every infinite cardinal $\alpha ,{\alpha ^{{\aleph _0}}} = \alpha$ if and only if there is a complete Boolean algebra $B$ s.t. card $B = \alpha$ (see [3, Theorem 25.4]). Recently, Comfort and Hager proved [1] that, for every infinite $\sigma$-complete Boolean algebra $B,{(\operatorname {card} B)^{{\aleph _0}}} = \operatorname {card} B$. We extend this result to the class of homomorphic images of $\sigma$-complete algebras, following closely Comfort’s and Hager’s proof. As a corollary, an improvement of Shelah’s theorem on the cardinality of ultraproducts of finite sets [2] is derived (Theorem 2).$^{1}$
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 171-175
  • MSC: Primary 06A40; Secondary 02H05
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0376475-9
  • MathSciNet review: 0376475