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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The minimal normal filter on $ P\sb{\kappa }\lambda $


Author: Donna M. Carr
Journal: Proc. Amer. Math. Soc. 86 (1982), 316-320
MSC: Primary 03E05; Secondary 04A20
MathSciNet review: 667297
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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$


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DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0667297-3
PII: S 0002-9939(1982)0667297-3
Article copyright: © Copyright 1982 American Mathematical Society