The minimal normal filter on $P_{\kappa }\lambda$
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- by Donna M. Carr
- Proc. Amer. Math. Soc. 86 (1982), 316-320
- DOI: https://doi.org/10.1090/S0002-9939-1982-0667297-3
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Abstract:
Let $\kappa$ be an uncountable regular cardinal, let $C{F_\kappa }$ be the cub filter on $\kappa$ and let $FS{F_\kappa }$ be the filter generated by $\{ \{ \beta < \kappa :\beta > \alpha \} :\alpha < \kappa \}$. It is well known that $C{F_k}$ is normal, that $C{F_\kappa } = \Delta FS{F_\kappa }$ and hence that every normal filter on $\kappa$ extends $C{F_\kappa }$. Jech extended some of these results to the context of ${P_\kappa }\lambda$. Let $\lambda$ be a cardinal $\geqslant \kappa$ and let $C{F_{\kappa \lambda }}$ denote the cub filter on ${P_{\kappa \lambda }}$ as defined by Jech; he showed that $C{F_{\kappa \lambda }}$ is normal and that every normal ultrafilter on ${P_{\kappa \lambda }}$ extends $C{F_{\kappa \lambda }}$. In this paper we extend these results further. In particular, we show that ${F_{\kappa \lambda }} = \Delta \Delta FS{F_{\kappa \lambda }}$ where $FS{F_{\kappa \lambda }}$ is the filter generated by $\{ \{ y \in {P_\kappa }\lambda :x \subset y\} :x \in {P_\kappa }\lambda \}$, and that every normal filter on ${P_\kappa }\lambda$ extends $CF{P_{\kappa \lambda }}$. Finally, we show that for any $\lambda \geqslant \kappa$ and any ideal $I$ on ${P_\kappa }\lambda ,\nabla \nabla \nabla I = \nabla \nabla I$References
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- Thomas J. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1972/73), 165–198. MR 325397, DOI 10.1016/0003-4843(73)90014-4 —, unpublished, 1978.
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Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 86 (1982), 316-320
- MSC: Primary 03E05; Secondary 04A20
- DOI: https://doi.org/10.1090/S0002-9939-1982-0667297-3
- MathSciNet review: 667297