Abstract

We strengthen a theorem of Gitik and Shelah [6] by showing that if κ is either weakly inaccessible or the successor of a singular cardinal and S is a stationary subset of κ such that $NS_{\kappa} \upharpoonright S$ is saturated then $\kappa \S$ is fat. Using this theorem we derive some results about the existence of fat stationary sets. We then strengthen some results due to Baumgartner and Taylor [2], showing in particular that if I is a $\lambda^{+++}-saturated$ normal ideal on $P_{\kappa} \lambda$ then the conditions of being $\lambda^{+}-preserving$ , weakly presaturated, and presaturated are equivalent for I