## Splitting $${\mathcal P}_\kappa \lambda$$ into maximally many stationary sets.(English)Zbl 0955.03047

Let $$\kappa$$ be a regular uncountable cardinal and let $$\lambda$$ be a cardinal, $$\lambda > \kappa$$. In this paper the author shows that $${\mathcal P}_{\kappa}\lambda = \{x\subseteq\lambda : |x|<\kappa\}$$ can be decomposed into $$\lambda^{\omega}$$ pairwise disjoint stationary sets, and when $$\text{ cf} \lambda < \kappa$$ then $${\mathcal P}_{\kappa}\lambda$$ can be decomposed into $$\lambda^+$$ pairwise disjoint stationary sets.

### MSC:

 300000 Other combinatorial set theory
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### References:

 [1] Y. Abe,Strongly normal ideals on P {$$\kappa$$} {$$\lambda$$} and the Sup-function, Topology and its Applications74 (1996), 97–107. · Zbl 0870.03020 [2] U. Abraham and S. Shelah,Forcing closed unbounded sets, Journal of Symbolic Logic48 (1983), 643–657. · Zbl 0568.03024 [3] J. Baumgartner,On the size of the closed unbounded sets, Annals of Pure and Applied Logic54 (1991), 195–227. · Zbl 0746.03040 [4] J. Baumgartner and A. Taylor,Saturation properties of ideals in generic extensions. I, Transactions of the American Mathematical Society270 (1982), 557–574. · Zbl 0485.03022 [5] M. Burke and M. Magidor,Shelah’s pcf theory and its applications, Annals of Pure and Applied Logic50 (1990), 207–254. · Zbl 0713.03024 [6] D. R. Burke and Y. Matsubara,The extent of strength in the club filters, Israel Journal of Mathematics114 (1999), 253–263 (this volume). · Zbl 0946.03056 [7] H.-D. Donder and P. Matet,Two cardinal versions of diamond, Israel Journal of Mathematics83 (1993), 1–43. · Zbl 0798.03047 [8] M. Foreman and M. Magidor,Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on P {$$\kappa$$}({$$\lambda$$}), Preprint, 1998. · Zbl 1017.03022 [9] M. Gitik,Nonsplitting subset of P {$$\kappa$$}({$$\kappa$$} +, Journal of Symbolic Logic50 (1985), 881–894. · Zbl 0601.03021 [10] Y. Hirata,Nonsaturation of the club filter on P {$$\kappa$$}{$$\lambda$$}, Master’s Thesis at University of Tsukuba, 1997. [11] A. Kanamori,The Higher Infinite, Springer, Berlin, 1994. · Zbl 0813.03034 [12] M. Kojman,The A, B, C of pcf: a companion to pcf theory, part I, Preprint, 1995. [13] M. Magidor,Representing sets of ordinals as countable unions of sets in the core model, Transactions of the American Mathematical Society317 (1990), 91–126. · Zbl 0714.03045 [14] Y. Matsubara,Menas’ conjecture and generic ultrapowers, Annals of Pure and Applied Logic36 (1987), 225–234. · Zbl 0632.03043 [15] T. Menas,On strong compactness and supercompactness, Annals of Mathematical Logic7 (1974), 327–359. · Zbl 0299.02084 [16] M. Rubin and S. Shelah,Combinatorial problems on trees: partitions, {$$\Delta$$}-systems and large free subsets, Annals of Pure and Applied Logic33 (1987), 43–81. · Zbl 0654.04002 [17] S. Shelah, {$$\omega$$}+1 has a Jonsson algebra, inCardinal Arithmetic, Oxford University Press, Oxford, 1994, pp. 34–116. [18] R. Solovay,Real-valued measurable cardinals, inAxiomatic Set Theory (D. Scott, ed.), Proceedings of Symposia in Pure Mathematics, Vol. 13, part 1, American Mathematical Society, Providence, 1971, pp. 397–428. [19] S. Todorčević,Partitioning pairs of countable sets, Proceedings of the American Mathematical Society111 (1991), 841–844. · Zbl 0722.03036 [20] S. Todorčević,Coding reals by sets of ordinals, Lectures at Nagoya University, 1994.
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