On shattering, splitting and reaping partitions

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In this article we investigate the dual-shattering cardinal H, the dual-splitting 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(omega) modulo finite, 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 omega, we show that add(J)=cov(J)=H. With this result we can show that H > omega₁ 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) is less than or equal to S (where M is the ideal of the meager sets). For the dual-reaping cardinal R we get p is less than or equal to R, which is less than or equal to r (where p is the pseudo-intersection number) and for a modified dual-reaping number R' we get that R' is less than or equal to d (where d is the dominating number). As a consistency result we get R < cov(M).