Distributivity and Stationary Reflections

In this paper, we present some relations between generalized distributivity of quotient algebras and Mahlo operations, and show that the distributivity implies some variants of stationary relections. The notion of generalized distributivity appeared in [3] for the first time. In that paper, we began investigating the relations between the notion and other fundamental principles in set theory, especially in the area called large cardinal axioms. For example, we showed in [3] that: Let I be any non-trivial κ-complete ideal on a regular uncountable cardinal κ and λ any cardinal with 2 ≤ λ < κ. Then 〈κ, (2; λ)〉-distributivity of the quotient ℘(κ)/I is equivalent to saying that there exists a non-trivial κ-complete prime ideal extending I (cf. Theorem 5). Then, it is interesting to discover the strength of distributivity in other forms, say, 〈κ, (μ; λ)〉-distributivity with λ ≥ κ. In this paper, we shall show that some versions imply some variants of stationary reflections. We shall organize our paper as follows: In §1, we shall introduce notations and terminologies, and several definitions of distributivity will be given. In §2, we shall investigate distributive conditions which imply some closedness of the Mahlo operation. Our main result in this section is the following: Theorem A. Let σ be any function of S into ℘(T ) such that for a, b in S, μa = |℘(σ(a))| < κ and if a ≺S b, then σ(a) ⊆ σ(b). Assume that I is a ≺S-fine κ-complete ≺S-normal 〈3, I 0 , (H ; μ)〉-distributive ideal on S, where H is the set ({0} × S) ∪ ({1} × T ) and μ = sup({|T |} ∪ {μa : a ∈ S }). Moreover, we assume that R = {a ∈ S : cf≺T (σ(a)) > א0 } has positive Imeasure, {a ∈ S : t ∈ σ(a) } has I-measure one for each t ∈ T and if g is a function on A ∈ I with g(a) ∈ σ(a), then there exists a subset B of A of positive I-measure such that gdB is constant. Then if X is a ≺T -stationary subset of T , R−Mσ(X) has I-measure zero. This theorem yields several corollaries. As applications of distributivity, we shall introduce those corollaries in §3. That is, the following will be shown. Received by the editors November 19, 1996 and, in revised form, January 5, 1998. 1991 Mathematics Subject Classification. Primary 03E55.