$μ$-Clubs OF $P_κ(λ)$ : Paradise on earth

Abstract

If $V = L$, and $μ$, $κ$ and $λ$ are three infinite cardinals with $μ= {\rm cf} (μ) < κ= {\rm cf}(κ) \leq λ$, then, as shown in \cite{Heaven}, the $μ$-club filters on $P_κ(λ)$ and $P_κ(λ^{< κ})$ are isomorphic if and only if ${\rm cf} (λ) \not= μ$. Now in $L$, $λ^{< κ}$ equals $u (κ, λ)$ (the least size of a cofinal subset in $(P_κ(λ), \subseteq)$) equals $λ$ if ${\rm cf} (λ) \geq κ$, and $λ^+$ otherwise. We show that, in ZFC, there are many triples $(μ, κ, λ)$ for which ($u (κ, λ) > λ$ and) the $μ$-club filters on $P_κ(λ)$ and $P_κ(u (κ, λ))$ are isomorphic.