On Shattering, Splitting and Reaping Partitions

In this article we investigate the dual-shattering cardinal H, the dualsplitting cardinal S and the dual-reaping cardinal R, which are dualizations of the well-known cardinals h (the shattering cardinal, also known as the distributivity number of P(ω)/fin), s (the splitting number) and r (the reaping number). Using some properties of the ideal J of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of ω, we show that add(J) = cov(J) = H. With this result we can show that H > ω1 is consistent with ZFC and as a corollary we get the relative consistency of H > t, where t is the tower number. Concerning S we show that cov(M) ≤ S (whereM is the ideal of the meager sets). For the dualreaping cardinal R we get p ≤ R ≤ r (where p is the pseudo-intersection number) and for a modified dual-reaping number R′ we get R′ ≤ d (where d is the dominating number). As a consistency result we get R < cov(M). 1 The set of partitions A partial partition X (of ω) consisting of pairwise disjoint, nonempty sets, such that dom(X) := ⋃ X ⊆ ω. The elements of a partial partition X are called the blocks of X and Min(X) denotes the set of the least elements of the blocks ofX. If dom(X) = ω, then X is called a partition. {ω} is the partition such that each block is a singleton and {{ω}} is the partition containing only one block. The set of all partitions containing infinitely (resp. finitely) many blocks is denoted by (ω)ω (resp. (ω)<ω). By (ω)ω we denote the set of all infinite partitions such that at least one block is infinite. The set of all partial partitions with dom(X) ∈ ω is denoted by (IIN). Let X1, X2 be two partial partitions. We say that X1 is coarser than X2, or that X2 is finer than X1, and write X1 v X2 if for all blocks b ∈ X1 the set b ∩ dom(X2) is the union of some sets bi ∩ dom(X1), where each bi is a block of X2. (Note that if X1 is coarser than X2, then X1 is in a natural way also contained in X2.) Let X1 uX2 denotes the finest partial partition which is coarser than X1 and X2 such that dom(X1 u X2) = dom(X1) ∪ dom(X2). Similarly X1 tX2 denotes the coarsest partial partition which is finer than X1 and X2 such that dom(X1 tX2) = dom(X1) ∪ dom(X2). The author wishes to thank the Swiss National Science Foundation for supporting him.