Large cardinal properties of ideals

For X？P_κλ，if|{t∈X：sup(T)=δ}|&lt；2^&lt；|δ|&gt；holds for everyδ&lt；λ，then we say that X is skinny.Let NS_&lt；κλ&gt；denote the non-stationary ideal over P_κλ。For X？P_κλdefine the ideal NS_&lt；κλ&gt；|X as follows；\Y？P_κλ(Y∈NS_&lt；κλ&gt；|X×X^Y∈NS_&lt；κλ&gt；).Y.Matsubara proved that if NS_&lt；κλ&gt；|X is precipitous then X has a skinny stationary subset of X·Previously Y.Matsubara and S.Shelah proved that ifλis a strong limit singular cardinal，then there is no skinny stationary subset of P_κλ.Therefore we can conclude that ifλis a strong limit singular cardinal，then NS_&lt；κλ&gt；|X cannot be precipitous for every X？P_κλ.We also proved that assuming GCH belowλthe exence of the precipitous&lt；Using this we can prove that under GCH the precipitousness of NS_&lt；κλ&gt；implies the diamond principle on every stationary subset of{α&lt；λ|cf(α)&lt；κ}。Therefore under GCH，the precipitousness of NS_&lt；κλ&gt；implies the ideal NS_λ|A cannot be saturated for every A.{α&lt；λ|cf(α)&lt；κ}。Here NS_λdenotes the non-stationary ideal overλand NS_λ|A denotes the ideal overλgenerated by NS_λandλ-A·

For X⊆P_κλ, if |{t∈X:sup(t)= δ}|<2^<|δ|> holds for every δ<λ, then we say that X is skinny. Let NS_<κλ> denote the non-stationary ideal over P_κλ. For X⊆P_κλ define the ideal NS_<κλ>|X as follows ; ∀Y⊆P_κλ(Y∈NS_<κλ>|X⇔X∩Y∈NS_<κλ>).Y.Matsubara proved that if NS_<κλ>|X is precipitous then X has a skinny stationary subset of X・Previously Y.Matsubara and S.Shelah proved that if λ is a strong limit singular cardinal, then there is no skinny stationary subset of P_κλ.Therefore we can conclude that if λ is a strong limit singular cardinal, then NS_<κλ>|X cannot be precipitous for every X⊆P_κλ.We also proved that assuming GCH below λ the existence of skinny stationary subset of P_κλ is equivalent to the diamond principle on {α<λ|cf(α)<κ}. Using this we can prove that under GCH the precipitousness of NS_<κλ> implies the diamond principle on every stationary subset of {α<λ|cf(α)<κ}. Therefore under GCH, the precipitousness of NS_<κλ> implies the ideal NS_λ|A cannot be saturated for every A⊆{α<λ|cf(α)<κ}. Here NS_λ denotes the non-stationary ideal over λ and NS_λ|A denotes the ideal over λ generated by NS_λ and λ-A・