The minimal normal $\mu$-complete filter on $P_ \kappa \lambda$

Abstract:

We introduce the closed (resp. strongly closed) $\mu$-unbounded filter $CF_{\kappa \lambda }^\mu$ (resp. $SCF_{\kappa \lambda }^\mu$) for a regular cardinal $\mu \leq \kappa$, which is a $\mu$-complete analog of the closed (resp. strongly closed) unbounded filter on ${P_\kappa }\lambda$. We give their simple characterizations for the case $\mu = {\omega _1}$. We also study generators of $CF_{\kappa \lambda }^\nu$ (resp. $SCF_{\kappa \lambda }^\nu$) over $CF_{\kappa \lambda }^\mu$ (resp. $SCF_{\kappa \lambda }^\mu$) for the case $\omega < \mu < \nu$.