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

Logic 2025-09-30 v3

Abstract

Denote by NA\mathcal{NA} and MA\mathcal{MA} the ideals of null-additive and meager-additive subsets of~2ω2^\omega, respectively. We prove in ZFC that add(NA)=non(NA)\mathrm{add}(\mathcal{NA})=\mathrm{non}(\mathcal{NA}) and introduce a new (Polish) relational system to reformulate Bartoszy\'nski's and Judah's characterization of the uniformity of MA\mathcal{MA}, which is helpful to understand the combinatorics of MA\mathcal{MA} and to prove consistency results. As for the latter, we prove that cov(MA)<c\mathrm{cov}(\mathcal{MA})<\mathfrak{c} (even cov(MA)<non(N)\mathrm{cov}(\mathcal{MA})<\mathrm{non}(\mathcal{N})) is consistent with ZFC, as well as several constellations of Cicho\'n's diagram with non(NA)\mathrm{non}(\mathcal{NA}), non(MA)\mathrm{non}(\mathcal{MA}) and add(SN)\mathrm{add}(\mathcal{SN}), which include non(NA)<b<non(MA)\mathrm{non}(\mathcal{NA})<\mathfrak{b}< \mathrm{non}(\mathcal{MA}) and b<add(SN)<cov(M)<d=c\mathfrak{b}< \mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}=\mathfrak{c}.

Keywords

Cite

@article{arxiv.2401.15364,
  title  = {Uniformity numbers of the null-additive and meager-additive ideals},
  author = {Miguel A. Cardona and Diego A. Mejía and Ismael E. Rivera-Madrid},
  journal= {arXiv preprint arXiv:2401.15364},
  year   = {2025}
}

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