$μ$-Clubs of $P_κ(λ)$ : Paradise in heaven

Abstract

Let $μ< κ< λ$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u (κ, λ)$ is regular, where $u (κ, λ)$ denotes the least size of a cofinal subset in $(P_κ(λ), \subseteq)$, and cf$(λ) \not= μ$, then (a) the $μ$-club filters on $P_κ(λ)$ and $P_κ(u (κ, λ))$ are isomorphic, and (b) the ideal dual to the $μ$-club filter on $P_κ(λ)$ (and hence the restriction of the nonstationary ideal on $P_κ(λ)$ to sets of uniform cofinality $μ$) is not $I_{κ, λ}$-$\frak{b}_{u (κ, λ)}$-saturated.