abstract
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On shattering, splitting and reaping partitions

Lorenz Halbeisen

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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* > omega_1 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*).