Yet another ideal version of the bounding number (2307.16017v1)

Abstract: 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 $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}{\mathcal{I}} \times \mathcal{D}{\mathcal{I}})) =\aleph_1$.