$\mu$-Clubs OF $P_{\kappa }(\lambda )$ : Paradise on earth

Abstract

If $V=L$, and $\mu$, $\kappa$ and $\lambda$ are three infinite cardinals with $\mu =\mathrm{cf}(\mu )<\kappa =\mathrm{cf}(\kappa )\le \lambda$, then, as shown in \cite{Heaven}, the $\mu$-club filters on $P_{\kappa }(\lambda )$ and $P_{\kappa }({\lambda }^{<\kappa })$ are isomorphic if and only if $\mathrm{cf}(\lambda )\ne \mu$. Now in $L$, ${\lambda }^{<\kappa }$ equals $u(\kappa ,\lambda )$ (the least size of a cofinal subset in $(P_{\kappa }(\lambda ),\subseteq )$) equals $\lambda$ if $\mathrm{cf}(\lambda )\ge \kappa$, and ${\lambda }^{+}$ otherwise. We show that, in ZFC, there are many triples $(\mu ,\kappa ,\lambda )$ for which ($u(\kappa ,\lambda )>\lambda$ and) the $\mu$-club filters on $P_{\kappa }(\lambda )$ and $P_{\kappa }(u(\kappa ,\lambda ))$ are isomorphic.