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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Homomorphic images of $ \sigma $-complete Boolean algebras


Author: Sabine Koppelberg
Journal: Proc. Amer. Math. Soc. 51 (1975), 171-175
MSC: Primary 06A40; Secondary 02H05
MathSciNet review: 0376475
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0376475-9
PII: S 0002-9939(1975)0376475-9
Article copyright: © Copyright 1975 American Mathematical Society