Large cardinal properties of ideals

for x⊆p _κ颗心,if | {t∈x: sup (t) =当}| < 2 ^ < |当| > holds for every当<颗心,那么我们说X是skinny. Let ns_ <κλ> denote the non-stationary ideal over p_ κλ。For X⊆p_ κλdefine the ideal ns_ <κλ>|X as follows；∀y⊆p _κ颗心(y∈ns _<κ颗心>| x←→x同样y∈ns _<κ颗心>). y.matsubara proved that if ns _<κ颗心>| x is precipitous then x has askinny stationary subset of X Previously Y.Matsubara and S.Shelah proved that if λ is a strong limitsingular cardinal,then there is no skinny stationary subset of p_κ λ.Therefore we can conclude that if λ is a stronglimit单一cardinalthenns_ <κλ>|X cannot be precipitous for every X⊆p_ κλ。我们也proved that assuming GCH below λ theexistence 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 thediamond 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 everyA⊆{α<λ|cf(α)<κ}. Here ns_λ denotes the non-stationary ideal over λ and ns_λ |A denotes the ideal overgenerated by ns_λ and λ-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・