Splitting stationary sets in P_κλ for λ with small cofinality

Abstract

Abstract. For a regular uncountable cardinal κ and a cardinal λ with cf(λ) < κ < λ, we investigate the consistency strength of the existence of a stationary set in Pκλ which cannot be split into λ + many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in Pκλ such that every stationary subset of S can be split into λ + many pairwise disjoint stationary subsets.

Menas’s Conjecture Revisited

“…If κ is weakly inaccessible, cf( ) < κ, and J is a normal, fine, weakly -saturated ideal on P κ ( ), then it gives us that the set of all b ∈ P κ ( ) such that (|b| > |b ∩ | for every cardinal with κ ≤ < , and hence) cf(|b|) = cf( ) lies in J * . In the special case when = κ + for some regular cardinal < κ, we can even conclude that {b ∈ P κ ( ) : |b| = |b ∩ κ| + } ∈ J * , since it is simple to see that for any cardinal with κ ≤ < , {b ∈ P κ ( ) : For normal ideals there is the following related result of Usuba (see the proof of Proposition 6.1 in [59]) who proved it using generic embeddings.…”

“…Fact 8.2 [59]. Suppose that is regular, and let J be a normal, fine ideal on P κ ( ) such that T ∈ J + , where T is the set of all x ∈ P κ ( ) such that (sup x is an infinite limit ordinal and) x is not stationary in sup x.…”

Weak saturation of ideals onPκ(λ)

“…Using different methods, Usuba [20] independently established the stronger result that if cf(λ) < κ and J is a normal, weakly λ + -saturated ideal on P κ (λ), then {b ∈ P κ (λ) : cf(|b|) = cf(λ)} ∈ J.…”

New combinatorial principle on singular cardinals and normal ideals

“…Cf. [16] for this topic. On the other hand, if j"λ + is in the generic ultrapower by the ideal (the corresponding generic ultrapower is not necessarily well-founded), then we can take a pseudo-diagonal intersection of λ + many measure one sets, and we can conclude that λ + many non-splitting of the stationary set yields the λ + -saturation property.…”