More on yet another ideal version of the bounding number

Abstract

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065--1092]. For an ideal $\mathcal{I}$ on $\omega$ we denote $\mathcal{D}_{\mathcal{I}}=\{f\in\omega^\omega: f^{-1}[\{n\}]\in\mathcal{I} \text{ for every $n\in \omega$}\}$ and write $f\leq_{\mathcal{I}} g$ if $\{n\in\omega:f(n)>g(n)\}\in\mathcal{I}$, where $f,g\in\omega^\omega$. We study the cardinal numbers $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))$ describing the smallest sizes of subsets of $\mathcal{D}_{\mathcal{I}}$ that are unbounded from below with respect to $\leq_{\mathcal{I}}$. In particular, we examine the relationships of $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))$ with the dominating number $\mathfrak{d}$. We show that, consistently, $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))>\mathfrak{d}$ for some ideal $\mathcal{I}$, however $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))\leq\mathfrak{d}$ for all analytic ideals $\mathcal{I}$. Moreover, we give example of a Borel ideal with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))=add(\mathcal{M})$.