The minimal normal 𝜇-complete filter on 𝑃_{𝜅}𝜆

Abstract

We introduce the closed (resp. strongly closed) μ \mu -unbounded filter C F κ λ μ CF_{\kappa \lambda }^\mu (resp. S C F κ λ μ SCF_{\kappa \lambda }^\mu ) for a regular cardinal μ ≤ κ \mu \leq \kappa , which is a μ \mu -complete analog of the closed (resp. strongly closed) unbounded filter on P κ λ {P_\kappa }\lambda . We give their simple characterizations for the case μ = ω 1 \mu = {\omega _1} . We also study generators of C F κ λ ν CF_{\kappa \lambda }^\nu (resp. S C F κ λ ν SCF_{\kappa \lambda }^\nu ) over C F κ λ μ CF_{\kappa \lambda }^\mu (resp. S C F κ λ μ SCF_{\kappa \lambda }^\mu ) for the case ω > μ > ν \omega > \mu > \nu .