More on yet another ideal version of the bounding number

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065--1092]. For an ideal I on ω we denote DI=∈ω^ω: f⁻¹[\] for everyn∈ ω\ and write f≤I g if ∈ω:f(n)>g(n)\, where f,g∈ω^ω. We study the cardinal numbers b(≥I∩ (I × I)) describing the smallest sizes of subsets of DI that are unbounded from below with respect to ≤I. In particular, we examine the relationships of b(≥I∩ (I × I)) with the dominating number d. We show that, consistently, b(≥I∩ (I × I))>d for some ideal I, however b(≥I∩ (I × I)) for all analytic ideals I. Moreover, we give example of a Borel ideal with b(≥I∩ (I × I))=add(M).