More on yet another ideal version of the bounding number

TL;DR

The paper investigates the bounding number associated with an ideal $\mathcal{I}$ on $\omega$, denoted $\mathfrak{b}(\mathcal{I})$, via the families $\mathcal{D}_{\mathcal{I}}$ and the relation $\leq_{\mathcal{I}}$. It establishes that $\mathfrak{b}(\mathcal{I})$ can exceed the dominating number $\mathfrak{d}$ for some $\mathcal{I}$, but for analytic ideals one always has $\mathfrak{b}(\mathcal{I})\leq\mathfrak{d}$; it also shows a Borel example with $\mathfrak{b}(\mathcal{I})=\mathrm{add}(\mathcal{M})$ and develops a comprehensive toolkit (topological representations, Katětov order, and Debs–Saint-Raymond methods) to compare $\mathfrak{b}(\mathcal{I})$ with $\mathfrak{d}$. A major contribution is a general construction yielding $\mathfrak{b}(\mathcal{I})>\kappa$ for suitable $\kappa$, plus a forcing argument demonstrating the consistency of $\mathfrak{b}(\mathcal{I})>\mathfrak{d}$ in certain models. The results clarify how the structure of an ideal (analytic, topological representation, etc.) constrains the related bounding number and its relation to the dominating number, with implications for ideal-QN-spaces and related combinatorics.

Abstract

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065--1092]. For an ideal $\mathcal{I}$ on $ω$ we denote $\mathcal{D}_{\mathcal{I}}=\{f\inω^ω: f^{-1}[\{n\}]\in\mathcal{I} \text{ for every $n\in ω$}\}$ and write $f\leq_{\mathcal{I}} g$ if $\{n\inω:f(n)>g(n)\}\in\mathcal{I}$, where $f,g\inω^ω$. 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 $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))=add(\mathcal{M})$.

Theorems & Definitions (29)

• Remark 2.1
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Definition 3.4
• Proposition 3.5
• proof
• Theorem 3.6