The minimal normal filter on $P_{\kappa }\lambda$

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 ${P_\kappa }\lambda ,\nabla \nabla \nabla I = \nabla \nabla I$