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

Abstract

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 $\lambda \geq \kappa$ 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 $\mathcal{P}_\kappa \lambda$.