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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Complete quotient Boolean algebras
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by Akihiro Kanamori and Saharon Shelah PDF
Trans. Amer. Math. Soc. 347 (1995), 1963-1979 Request permission

Abstract:

For $I$ a proper, countably complete ideal on the power set $\mathcal {P}(x)$ for some set $X$, can the quotient Boolean algebra $\mathcal {P}(X)/I$ be complete? We first show that, if the cardinality of $X$ is at least ${\omega _3}$, then having completeness implies the existence of an inner model with a measurable cardinal. A well-known situation that entails completeness is when the ideal $I$ is a (nontrivial) ideal over a cardinal $\kappa$ which is ${\kappa ^ + }$-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over $\kappa = {\omega _1}$ relative to the existence of a Woodin cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large cardinal hypothesis, the consistency of: ${2^{{\omega _1}}} = {\omega _3}$ and there is an ideal ideal $I$ over ${\omega _1}$ such that $\mathcal {P}({\omega _1})/I$ is complete. (The cardinality assertion implies that there is no ideal over ${\omega _1}$ which is ${\omega _2}$-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1963-1979
  • MSC: Primary 03E35; Secondary 03E40, 03E55, 06E05
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1282888-0
  • MathSciNet review: 1282888