Homomorphic images of -complete Boolean algebras
Proc. Amer. Math. Soc. 51 (1975), 171-175
Primary 06A40; Secondary 02H05
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Abstract: It is a well-known theorem of R. S. Pierce that, for every infinite cardinal if and only if there is a complete Boolean algebra s.t. card (see [3, Theorem 25.4]). Recently, Comfort and Hager proved  that, for every infinite -complete Boolean algebra . We extend this result to the class of homomorphic images of -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  is derived (Theorem 2).
W. Comfort and Anthony
W. Hager, Cardinality of 𝔨-complete Boolean algebras,
Pacific J. Math. 40 (1972), 541–545. MR 0307997
Shelah, On the cardinality of ultraproduct of finite sets, J.
Symbolic Logic 35 (1970), 83–84. MR 0325388
R. Sikorski, Boolean algebras, 2nd ed., Springer-Verlag, Berlin and New York, 1964.
- W. W. Comfort and A. W. Hager, Cardinality of -complete Boolean algebras, Pacific J. Math. 40 (1972), 541-545. MR 46 #7112. MR 0307997 (46:7112)
- S. Shelah, On the cardinality of ultraproducts of finite sets, J. Symbolic Logic 35 (1970), 83-84. MR 0325388 (48:3735)
- R. Sikorski, Boolean algebras, 2nd ed., Springer-Verlag, Berlin and New York, 1964.
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