Ideals on P_κλ considered from the point of view of the relation with large cardinals axioms and the generalized continuum hypothesis

(1)我们证明了|Pκλ| =λ，如果κ是轻度λ不可描述的，且cf（λ）<$κ。(2)我们证明了Pκλ上的俱乐部滤波器在反射原理下是弱预饱和的。(3)我们证明了n是温和的。不可描述的Pκ（2^<λ<κ>）具有弱配分性质。(4)我们证明了如果（κ，λ]中没有基数是II^2_1-不可描述的，则Pκλ的每一个无效子集都具有分划性质. (5)从一个大基数模型出发，建立了一个简单的强迫模型，其中ω_1带有一个ω_2饱和理想。(6)本文将Shelah关于Pω_1λ（ω_，λ，）中的钻石原理的一个定理推广到Pκλ中：（a）当2^ω=2^κ且κ<λ时，ω_<κ，λ>成立;（B）在超紧基数模型的基础上，我们给出了一个强迫模型，其中2^ω=2^κ，而当κ< λ时，ω_<κ，λ>成立。(7)我们得到了理想I_f的共尾性、可加性数和覆盖数的一些独立结果。

(1) We proved that |Pκλ|=λ if κ is mildly λ ineffable and cf(λ)〓κ.(2) We showed the club filter on Pκλ is weakly presaturated under the Reflection principle.(3) We showed n is mildly A ineffable Pκ(2^<λ<κ>) has the weak partition property.(4) We proved every ineffbale subset of Pκλ has the partition property if no cardinal in (κ, λ] is II^2_1-indescribable.(5) Starting from a model of a huge cardinal we built a simple forcing model in which ω_1 carries an ω_2 saturated ideal.(6) We extended a theorem of Shelah on the diamond principle in Pω_1λ(◇ω_, λ,) to Pκλ as follows:(a) ◇_<κ, λ> holds if 2^ω=2^κ with κ<λ(b) Building upon a model of a supercompact cardinal, we gave a forcing model in which 2^ω=2^κ whereas holds whenever κ< λ.(7) We got some independence results for cofinality, additivity number, and covering number of the ideal I_f.