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The reaping and splitting numbers of nice ideals

Volume 134 / 2014

Rafał Filipów Colloquium Mathematicum 134 (2014), 179-192 MSC: Primary 03E17, 40A35; Secondary 03E50, 40A30, 03E75. DOI: 10.4064/cm134-2-3

Abstract

We examine the splitting number $\mathfrak {s}(\mathbf {B})$ and the reaping number $\mathfrak {r}(\mathbf {{B}})$ of quotient Boolean algebras $\mathbf {B}=\mathcal {P}(\omega )/\mathcal {I}$ where $\mathcal {I}$ is an $F_\sigma $ ideal or an analytic P-ideal. For instance we prove that under Martin's Axiom $\mathfrak {s}(\mathcal {P}(\omega )/\mathcal {I})=\mathfrak {c}$ for all $F_\sigma $ ideals $\mathcal {I}$ and for all analytic P-ideals $\mathcal {I}$ with the $\textrm {BW}$ property (and one cannot drop the $\textrm {BW}$ assumption). On the other hand under Martin's Axiom $\mathfrak {r}(\mathcal {P}(\omega )/\mathcal {I})=\mathfrak {c}$ for all $F_\sigma $ ideals and all analytic P-ideals $\mathcal {I}$ (in this case we do not need the $\textrm {BW}$ property). We also provide applications of these characteristics to the ideal convergence of sequences of real-valued functions defined on the reals.

Authors

  • Rafał FilipówInstitute of Mathematics
    University of Gdańsk
    Wita Stwosza 57
    80-952 Gdańsk, Poland
    e-mail

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