More on cardinal invariants of analytic P-ideals

Abstract

Given an ideal $I$ on $\omega$ let $a(I)$ ($\bar{a}(I)$) be minimum of the cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of $[\omega {]}^{\omega }$, and denote $b_I$ and$d_I$ the unbounding and dominating numbers of $({\omega }^{\omega },{\le }_I)$. We show that (1) $a(I)>omega$ if $I$ is a summable ideal; (2) $a(Z)=\omega$ and $\bar{a}(Z)\le a$ if $Z$ is a tall density ideal, (3) $b\le \bar{a}(I)$, and $b_I=b$ and $d_I=d$, for any analytic P-ideal $I$ on $\omega$. Given an analytic $P$-ideal $I$ we investigate the relationship between the Sack, the $I$-bounding, $I$-dominating and ${\omega }^{\omega }$-bounding properties of a given poset $P$. For example, for the density zero ideal $Z$ we can prove: (i) a poset $P$ is $Z$-bounding iff it has the Sacks property, (ii) if $P$ adds a slalom capturing all ground model reals then $P$ is $Z$-dominating.