On splitting stationary subsets of large cardinals

Abstract

AbstractLet κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ+-saturated, i.e., are there κ+ stationary subsets of κ with pairwise intersections nonstationary? Our first observation is:Theorem. NS isκ+-saturated iff for every normal ideal J on κ there is a stationary set A ⊆ κsuch that J = NS∣A = {X ⊆ κ: X ∩ A ∈ NS}.Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define “greatly Mahlo” cardinals and obtain the following:Theorem. If κ is greatly Mahlo then NS is notκ+-saturated.Theorem. If κ is ordinal Π11-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries aκ-saturated ideal, thenκis greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π11-indescribable cardinal.These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ+-saturated.