On $2$-stationarity of ${\mathcal{P}}_{\kappa }\lambda$

The notion of $n$-stationary subsets of ${\mathcal{P}}_{\kappa }\lambda$ for $n<\omega$ were introduced and studied by Cody, Lambie-Hanson & Zhang \cite{CLHZ} and Torres \cite{T}. They proved that if $\kappa$ is supercompact, then ${\mathcal{P}}_{\kappa }\lambda$ is $n$-stationary in itself for all cardinals $\lambda \ge \kappa$ and all $n<\omega$. In this paper we prove that strong compactness of $\kappa$ does not imply the $2$-stationarity of ${\mathcal{P}}_{\kappa }\lambda$ for cardinals $\lambda >\kappa$. We also discuss Menas’ Theorem for $1$-stationary and $2$-stationary subsets of ${\mathcal{P}}_{\kappa }\lambda$.