abstract
On shattering, splitting and reaping partitions
Lorenz Halbeisen
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).