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On a generalization of distributivity

In this paper, we generalize the notion of distributivity and consider some properties of distributive ideals, that is, ideals I such that the algebra P(κ)/I is distributive in our sense. Our notation and terminology is explained in §1, while the main results of this paper begin in §2. We shall show here some relations of the distributivity and the ideal theoretic partitions. In §3, we shall study the class of distributive ideals over κ whose existence is equivalent to the ineffability of κ, and other classes. Finally, in §4, we shall consider the equivalence of the Boolean prime ideal theorem and show that the existence of certain distributive ideals characterizes several large cardinals. As a byproduct, we can give a simple proof of Ketonen's theorem that κ is strongly compact if and only if for any regular cardinal λ ≥ κ there exists a nontrivial κ-complete prime ideal over λ.