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Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on \(P_x(\lambda)\). (English) Zbl 1017.03022

The authors investigate the saturation of the non-stationary ideal on \(P_\kappa(\lambda)\). The results of the paper can be summarized as: This ideal is not (\(\lambda^+\)-)saturated unless \(\kappa=\lambda=\aleph_1\). In some special cases stronger results are available. For example, if \(\lambda\) is singular then \(\text{NS}(P_{\aleph_1}(\lambda))\) is not \(\lambda^{\text{cf}\lambda}\)-saturated, and if \(\text{cf}\lambda<\kappa\geq\aleph_2\) then \(\text{NS}(P_\kappa(\lambda))\) is not \(\lambda^{++}\)-saturated (D. R. Burke and Y. Matsubara [Isr. J. Math. 114, 253-263 (1999; Zbl 0946.03056)] showed it is not \(\lambda^+\)-saturated).
The paper closes with a discussion of one of its main tools: sequences of mutually stationary sets. If \(K\) is a set of regular cardinals with supremum \(\delta\) and \(S_\kappa\subseteq\kappa\) for all \(\kappa\in K\) then \(\{S_\kappa:\kappa\in K\}\) is mutually stationary if every algebra on \(\delta\) has an elementary substructure \(N\) that satisfies the implication \((\kappa\in N\cap K) \Rightarrow (\sup(N\cap\kappa)\in S_\kappa)\).
Reviewer: K.P.Hart (Delft)

MSC:

03E05 Other combinatorial set theory
03E04 Ordered sets and their cofinalities; pcf theory
03E45 Inner models, including constructibility, ordinal definability, and core models
03E55 Large cardinals

Citations:

Zbl 0946.03056
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References:

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