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A note on a lemma of Shelah concerning stationary sets

Authors: Alan H. Mekler, Donald H. Pelletier and Alan D. Taylor
Journal: Proc. Amer. Math. Soc. 83 (1981), 764-768
MSC: Primary 04A20
MathSciNet review: 630051
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Abstract: Let $ \kappa $ be an infinite cardinal, let $ I$ be a nonprincipal ideal on $ \kappa $ and let $ {I^ + } = \{ X \subseteq \kappa :X \notin I\} $. $ S(I)$ is the following property of ideals: for every $ A \in {I^ + }$ and every pair of functions $ f,g$ from $ A$ into $ \kappa $ such that, for every $ \alpha \in A$, $ f(a) \ne g(\alpha )$, there exists a set $ B \subseteq A$ with $ B \in {I^ + }$ such that $ f''B \cap g''B = \emptyset $. We prove that $ S(I)$ holds for every weakly selective ideal $ I$ on any infinite cardinal $ \kappa $ (including $ \kappa = \omega $), and that $ S(I)$ holds for every $ \kappa $-complete ideal on $ \kappa $ iff $ \kappa $ is not strongly inaccessible.

References [Enhancements On Off] (What's this?)

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Keywords: Ideal, stationary sets, normal ideal, weakly selective ideal
Article copyright: © Copyright 1981 American Mathematical Society

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