The reaping and splitting numbers of nice ideals

Rafał Filipów¹

¹ Institute of Mathematics University of Gdańsk Wita Stwosza 57 80-952 Gdańsk, Poland

Colloquium Mathematicum, Tome 134 (2014) no. 2, pp. 179-192

Résumé

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.

Détail

DOI : 10.4064/cm134-2-3

Keywords: examine splitting number mathfrak mathbf reaping number mathfrak mathbf quotient boolean algebras mathbf mathcal omega mathcal where mathcal sigma ideal analytic p ideal instance prove under martins axiom mathfrak mathcal omega mathcal mathfrak sigma ideals mathcal analytic p ideals mathcal textrm property cannot drop textrm assumption other under martins axiom mathfrak mathcal omega mathcal mathfrak sigma ideals analytic p ideals mathcal textrm property provide applications these characteristics ideal convergence sequences real valued functions defined reals