Recall that a nonempty set ${\cal I}\subseteq {\cal P}(\omega)$ is a (set) ideal if
- $B\in{\cal I}$ and $A\subseteq B$ imply $A\in{\cal I}$, and
- $A,B \in {\cal I}$ implies $A\cup B\in {\cal I}$.
By $\text{Id}(\omega)$ we denote the set of all ideals on $\omega$. The Katetov ordering on $\text{Id}(\omega)$ is defined by $${\cal I} \leq_K {\cal J} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall I\in{\cal I}) f^{-1}(I)\in J .$$
It is easy to see that $\leq_K$ is reflexive and transitive, but not anti-symmetric. Set ${\cal I}\simeq_K {\cal J}$ if ${\cal I}\leq_K{\cal J}$ and ${\cal J}\leq_K{\cal I}$.
Questions.
1) What are the cardinalities of the equivalence classes $[{\cal I}]_{\simeq_K}$ for ${\cal I}\in\text{Id}(\omega)$?
2) What is the cardinality of the poset $\text{Id}(\omega)/\simeq_K$?