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Yet another ideal version of the bounding number. (English) Zbl 1504.03025

For an ideal \(\mathcal{I}\) on \(\omega\) let \(\mathcal{D}_\mathcal{I}= \{f\in\omega^\omega:f^{-1}[\{n\}]\in\mathcal{I}\) for all \(n\in\omega\}\) and for \(f,g\in\omega^\omega\) let \(f\le_\mathcal{I}g\) if \(\{n\in\omega:f(n)>g(n)\}\in\mathcal{I}\). The authors examine the bounding number \(\mathfrak{b} ({\ge_\mathcal{I}}\cap(\mathcal{D}_\mathcal{K}\times\mathcal{D}_\mathcal{J}))\) of the relation \({\ge_\mathcal{I}}\cap(\mathcal{K}\times\mathcal{J})\) for ideals \(\mathcal{I}\), \(\mathcal{J}\), \(\mathcal{K}\) on \(\omega\). They prove that for every ideal \(\mathcal{I}\) with the Baire property, \(\mathfrak{b} ({\ge_\mathcal{I}}\cap(\mathcal{D}_\text{Fin}\times\mathcal{D}_\text{Fin}))= \mathfrak{b}\) and for every coanalytic weak P-ideal \(\mathcal{I}\), \(\aleph_1\le \mathfrak{b} ({\ge_\mathcal{I}}\cap(\mathcal{D}_\mathcal{I}\times\mathcal{D}_\mathcal{I}))\le \mathfrak{b} ({\ge_\mathcal{I}}\cap(\mathcal{D}_\text{Fin}\times\mathcal{D}_\mathcal{I}))= \mathfrak{b}\). They give examples of \(\boldsymbol\Sigma^0_2\) ideals with \(\mathfrak{b} ({\ge_\mathcal{I}}\cap(\mathcal{D}_\mathcal{I}\times\mathcal{D}_\mathcal{I}))= \mathfrak{b}\) as well as examples of \(\boldsymbol\Sigma^0_2\) ideals with \(\mathfrak{b} ({\ge_\mathcal{I}}\cap(\mathcal{D}_\mathcal{I}\times\mathcal{D}_\mathcal{I}))= \aleph_1\). The article is a fairly detailed analysis of these cardinal invariants.

MSC:

03E05 Other combinatorial set theory
03E15 Descriptive set theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
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