Ideals and combinatorial principles

Abstract

It is well known that if σ is a strongly compact cardinal and λ a regular cardinal ≥ σ, then for every stationary subset X of {α < λ: cof (α) = ω} there is some β < λ such that X ⋂ β is stationary in β. In fact the existence of a uniform, countably complete ultrafilter over λ is sufficient to prove the same conclusion about stationary subsets of {α < λ: cof (α) = ω}. See [13] or [10]. By analyzing the proof of this theorem as presented in [10], we realized the same conclusion will follow from the existence of a certain ideal, not necessarily prime, on . Throughout we will assume that σ is a regular uncountable cardinal and use the word “ideal” to mean fine ideal.