## Splitting stationary sets in $$\mathcal P(\lambda )$$.(English)Zbl 1250.03077

Summary: Let $$A$$ be a non-empty set. A set $$S\subseteq \mathcal {P}(A)$$ is said to be stationary in $$\mathcal{P}(A)$$ if for every $$f: [A]^{<\omega } \rightarrow A$$ there exists $$x\in S$$ such that $$x\neq A$$ and $$f[x]^{<\omega } \subseteq x$$. In this paper we prove the following: For an uncountable cardinal $$\lambda$$ and a stationary set $$S$$ in $$\mathcal{P}(\lambda)$$, if there is a regular uncountable cardinal $$\kappa\leq \lambda$$ such that $$\{x \in S: x \cap \kappa \in \kappa\}$$ is stationary, then $$S$$ can be split into $$\kappa$$ disjoint stationary subsets.

### MSC:

 300000 Other combinatorial set theory 3e+55 Large cardinals

### Keywords:

stationary set; saturated ideal; pcf-theory
Full Text:

### References:

 [1] U. Abraham and M. Magidor Cardinal arithmetic , Handbook of set theory (Matthew Foreman and Akihiro Kanamori, editors), vol. II, Springer-Verlag, Berlin,2010, pp. 1149-1228. · Zbl 1198.03053 [2] D. Burke Splitting stationary subsets of $$\p(\chi)$$ , unpublished. · Zbl 1199.54065 [3] M. Foreman Potent axioms , Transactions of the American Mathematical Society , vol. 294(1986), no. 1, pp. 1-28. · Zbl 0614.03051 [4] —- Ideals and generic elementary embeddings , Handbook of set theory (Matthew Foreman and Akihiro Kanamori, editors), vol. II, Springer-Verlag, Berlin,2010, pp. 885-1148. · Zbl 1198.03050 [5] M. Gitik Nonsplitting subset of $$P_ \kappa(\kappa^ +)$$ , Journal of Symbolic Logic, vol. 50(1985), no. 4, pp. 881-894. · Zbl 0601.03021 [6] T. Jech Some combinatorial problems concerning uncountable cardinals , Annals of Mathematical Logic , vol. 5(1973), pp. 165-198. · Zbl 0262.02062 [7] A. Kanamori The higher infinite: Large cardinals in set theory from their beginnings , Perspective in Mathematical Logic, Springer-Verlag, Berlin,1994. · Zbl 0813.03034 [8] P. Larson The stationary tower. Notes on a course by W. Hugh Woodin , University Lecture Series, 32, American Mathematical Society,2004. · Zbl 1072.03031 [9] Y. Matsubara Consistency of Menas’ conjecture , Journal of the Mathematical Society of Japan , vol. 42(1990), no. 2, pp. 259-263. · Zbl 0704.03032 [10] S. Shelah Cardinal arithmetic , Oxford Logic Guides, 29, Oxford Science Publications,1994. · Zbl 0848.03025 [11] M. Shioya A saturated stationary subset of $$\mathcal{P}_\kappa \kappa^+$$ , Mathematical Research Letters , vol. 10(2003), no. 4, pp. 493-500. · Zbl 1044.03031 [12] R. Solovay Real valued measurable cardinals , Axiomatic set theory $$($$Proceedings of the symposium on Pure Mathematics, Vol XIII, Part I, University of California, Los Angeles, California, 1967$$)$$ , American Mathematical Society, Providence, R.I,1971, pp. 397-428.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.