## 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:

 300000 Other combinatorial set theory 30000 Ordered sets and their cofinalities; pcf theory 3e+45 Inner models, including constructibility, ordinal definability, and core models 3e+55 Large cardinals

Zbl 0946.03056
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