Large cardinal properties of ideals

对于X P_κλ，如果|{t∈X：sup（t）= δ}|<2>>| δ（δ）|把每个δ都保留下来<γ，然后我们说X很瘦。Let NS_<κλ> denote the non-stationary ideal over P_κλ。对于X μ P_κλ定义理想NS_<κλ>| X as follows ; Y P_κλ（Y∈ NS_<κλ>）|X X∩Y∈ NS_<κλ>）. Y. Matsubara提供，如果NS_<κλ>| X是精确的，然后X是X·先前Y. Matsubara和S. Shelah提供的，如果γ是一个强限制的单一基数，那么没有P_κλ的皮肤静态基数，所以我们可以将其结合起来，如果γ是一个强限制的单一基数，然后NS_<κλ>| X不可能对每一个X P_κλ都是精确的，我们也证明了假定GCH下面γ的存在P_κλ的皮肤静态子集与钻石原则相等。|cf（α）<κ}。使用这个我们可以证明，在GCH下NS_<γ>的精确度在{α<γ>的每一个静态子集上实现钻石原则|cf（α）<κ}。因此，在GCH下，NS_<κλ>的准确性得到了理想的NS_λ| A不能对每个A {α<λ| cf（α）<κ}。这里NS_λ拒绝使用λ和NS_λ的非静态理想|一个理想的由NS_λ和λ-A产生的拒绝

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・