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If $V = L$, and $\mu$, $\kappa$ and $\lambda$ are three infinite cardinalswith $\mu = {\rm cf} (\mu) < \kappa = {\rm cf}(\kappa) \leq \lambda$, then, asshown in \cite{Heaven}, the $\mu$-club filters on $P_\kappa (\lambda)$ and$P_\kappa (\lambda^{< \kappa})$ are isomorphic if and only if ${\rm cf}(\lambda) \not= \mu$. Now in $L$, $\lambda^{< \kappa}$ equals $u (\kappa,\lambda)$ (the least size of a cofinal subset in $(P_\kappa (\lambda),\subseteq)$) equals $\lambda$ if ${\rm cf} (\lambda) \geq \kappa$, 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.

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