Uniformity numbers of the null-additive and meager-additive ideals

Abstract

Denote by $\mathcal{N}\mathcal{A}$ and $\mathcal{M}\mathcal{A}$ the ideals of null-additive and meager-additive subsets of~$2^{\omega }$, respectively. We prove in ZFC that $\mathrm{add}(\mathcal{N}\mathcal{A})=\mathrm{non}(\mathcal{N}\mathcal{A})$ and introduce a new (Polish) relational system to reformulate Bartoszyński's and Judah's characterization of the uniformity of $\mathcal{M}\mathcal{A}$, which is helpful to understand the combinatorics of $\mathcal{M}\mathcal{A}$ and to prove consistency results. As for the latter, we prove that $\mathrm{cov}(\mathcal{M}\mathcal{A})<\mathfrak{c}$ (even $\mathrm{cov}(\mathcal{M}\mathcal{A})<\mathrm{non}(\mathcal{N})$) is consistent with ZFC, as well as several constellations of Cichoń's diagram with $\mathrm{non}(\mathcal{N}\mathcal{A})$, $\mathrm{non}(\mathcal{M}\mathcal{A})$ and $\mathrm{add}(\mathcal{S}\mathcal{N})$, which include $\mathrm{non}(\mathcal{N}\mathcal{A})<\mathfrak{b}<\mathrm{non}(\mathcal{M}\mathcal{A})$ and $\mathfrak{b}<\mathrm{add}(\mathcal{S}\mathcal{N})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}=\mathfrak{c}$.

Final version

References in corpus (1)

• More about the cofinality and the covering of the ideal of strong measure zero sets

Cited by in corpus (1)

• Directed schemes of ideals and cardinal characteristics, I: the meager additive ideal