  Let $\mathcal{I}$ be an ideal on $\omega$. For $f,g\in\omega^\omega$ we write
$f \leq_{\mathcal{I}} g$ if $f(n) \leq g(n)$ for all $n\in\omega\setminus A$
with some $A\in\mathcal{I}$. Moreover, we denote
$\mathcal{D}_{\mathcal{I}}=\{f\in\omega^\omega: f^{-1}[\{n\}]\in\mathcal{I}
\text{ for every $n\in \omega$}\}$ (in particular, $\mathcal{D}_{Fin}$ denotes
the family of all finite-to-one functions).
  We examine cardinal numbers $\mathfrak{b}(\geq_{\mathcal{I}}\cap
(\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))$ and
$\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{Fin}\times
\mathcal{D}_{Fin}))$ describing the smallest sizes of unbounded from below with
respect to the order $\leq_{\mathcal{I}}$ sets in $\mathcal{D}_{Fin}$ and
$\mathcal{D}_{\mathcal{I}}$, respectively. For a maximal ideal $\mathcal{I}$,
these cardinals were investigated by M. Canjar in connection with coinitial and
cofinal subsets of the ultrapowers.
  We show that $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{Fin} \times
\mathcal{D}_{Fin})) =\mathfrak{b}$ for all ideals $\mathcal{I}$ with the Baire
property and that $\aleph_1 \leq \mathfrak{b}(\geq_{\mathcal{I}}\cap
(\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}})) \leq\mathfrak{b}$
for all coanalytic weak P-ideals (this class contains all $\Pi^0_4$ ideals).
What is more, we give examples of Borel (even $\Sigma^0_2$) ideals
$\mathcal{I}$ with $\mathfrak{b}(\geq_{\mathcal{I}}\cap
(\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))=\mathfrak{b}$ as
well as with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}}
\times \mathcal{D}_{\mathcal{I}})) =\aleph_1$.

Logic