On Shattering,Splitting and Reaping Partitions

In this article we investigate the dual-shattering cardinal ?, the dual-splitting cardinal ?? and the dual-reaping cardinal ??, which are dualizations of the well-known cardinals ?? (the shattering cardinal, also known as the distributivity number of P(ω)/fin), s (the splitting number) and ?? (the reaping number). Using some properties of the ideal ?? of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of ω, we show that add(??) = cov(??) = ?. With this result we can show that ? > ω₁ is consistent with ZFC and as a corollary we get the relative consistency of ? > ?? t, where t is the tower number. Concerning ?? we show that cov(M) ? ?? ?? (where M is the ideal of the meager sets). For the dual-reaping cardinal ?? we get p ?? ? ?? ? ?? (where ?? is the pseudo-intersection number) and for a modified dual-reaping number ??′ we get ??′ ? ?? (where ?? is the dominating number). As a consistency result we get ?? < cov(??).