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Given an ideal $ℐ$ on $\omega$ let $𝔞\left(ℐ\right)$ ($\overline{𝔞}\left(ℐ\right)$) be minimum of the cardinalities of infinite (uncountable) maximal $ℐ$-almost disjoint subsets of ${\left[\omega \right]}^{\omega }$. We show that $𝔞\left({ℐ}_h\right)>\omega$ if ${ℐ}_h$ is a summable ideal; but $𝔞\left({𝒵}_{\overrightarrow{\mu }}\right)=\omega$ for any tall density ideal ${𝒵}_{\overrightarrow{\mu }}$ including the density zero ideal $𝒵$. On the other hand, you have $𝔟\le \overline{𝔞}\left(ℐ\right)$ for any analytic $P$-ideal $ℐ$, and $\overline{𝔞}\left({𝒵}_{\overrightarrow{\mu }}\right)\le 𝔞$ for each density ideal ${𝒵}_{\overrightarrow{\mu }}$. For each ideal $ℐ$ on $\omega$ denote ${𝔟}_{ℐ}$ and ${𝔡}_{ℐ}$ the unbounding and dominating numbers of $\langle {\omega }^{\omega },{\le }_{ℐ}\rangle$ where $f{\le }_{ℐ}g$ iff $\{n\in \omega :f(n)>g(n\left)\right\}\in ℐ$. We show that ${𝔟}_{ℐ}=𝔟$ and ${𝔡}_{ℐ}=𝔡$ for each analytic $P$-ideal $ℐ$. Given a Borel ideal $ℐ$ on...