Distributive ideals and partition relations

Extract

It is a theorem of Rowbottom [12] that if κ is measurable and I is a normal prime ideal on κ, then for each λ < κ,

In this paper a natural structural property of ideals, distributivity, is considered and shown to be related to this and other ideal theoretic partition relations.

The set theoretical terminology is standard (see [7]) and background results on the theory of ideals may be found in [5] and [8]. Throughout κ will denote an uncountable regular cardinal, and I a proper, nonprincipal, κ-complete ideal on κ. NS_κ is the ideal of nonstationary subsets of κ, and I_κ = {X ⊆ κ∣∣X∣<κ}. If A ∈ I⁺ (= P(κ) − I), then an I-partition of A is a maximal collection W ⊆, P(A) ∩ I⁺ so that X ∩ Y ∈ I whenever X, Y ∈ W, X ≠ Y. The I-partition W is said to be disjoint if distinct members of W are disjoint, and in this case, for denotes the unique member of W containing ξ. A sequence 〈W_α ∣ α < η} of I-partitions of A is said to be decreasing if whenever α < β < η and X ∈ W_β there is a Y ∈ W_α such that X ⊆ Y. (i.e., W_β refines W_α).