More on yet another ideal version of the bounding number

TL;DR

This paper investigates certain cardinal invariants related to ideals on natural numbers, exploring their relationships with the dominating number and providing examples with specific bounds, including for Borel ideals.

Contribution

It introduces new bounds for these invariants, compares them with the dominating number, and provides examples for various classes of ideals, including Borel and analytic ideals.

Findings

Consistently, the invariant can exceed the dominating number for some ideals.

For all analytic ideals, the invariant is at most the dominating number.

A Borel ideal example shows the invariant equals the additivity of the meager ideal.

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 $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))=add(\mathcal{M})$.