Around splitting and reaping for partitions of ω

Abstract

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \({\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}}\) and \({\mathfrak{s}_{d}\geq\mathfrak{b}}\) . We also show the consistency results \({\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}\) and \({\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}\) . To prove the consistency \({\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}\) and \({\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})}\) we introduce new cardinal invariants \({\mathfrak{r}_{pair}}\) and \({\mathfrak{s}_{pair}}\) . We also study the relation between \({\mathfrak{r}_{pair}, \mathfrak{s}_{pair}}\) and other cardinal invariants. We show that \({\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}}\) and \({\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})}\) .