Around splitting and reaping for partitions of \(\omega \)
Let \(((\omega)^\omega,\leq^*)\) be the structure of infinite partitions of \(\omega\) ordered by ``almost coarser''. The study of cardinal invariants of \(((\omega)^\omega,\leq^*)\) was initiated by \textit{P. Matet} in [J. Symb. Log. 51, 12--21 (1986; Zbl 0588.04011)], and they were investigated comprehensively by \textit{J. Cichoń, A. Krawczyk, B. Majcher-Iwanow} and \textit{B. Węglorz} in [J. Symb. Log. 65, No.~2, 959--968 (2000; Zbl 0968.03056)], \textit{L. Halbeisen} in [Math. Log. Q. 44, No.~1, 123--134 (1998; Zbl 0896.03036)], \textit{O. Spinas} in [Ann. Pure Appl. Logic 90, No.~1--3, 243--262 (1997; Zbl 0891.03019)] and \textit{J. Brendle} in [Math. Log. Q. 46, No.~2, 241--248 (2000; Zbl 0993.03063)]. Their relationship is displayed in the dual van Douwen diagram. In the article under review the author investigates the splitting number \({\mathfrak s}_d\) and the reaping number \({\mathfrak r}_d\) for the structure \(((\omega)^\omega,\leq^*)\) and proves some relations between these invariants and cardinal invariants in Cichoń's diagram. In particular, he shows that \({\mathfrak r}_d\leq\text{non}({\mathcal M}),\text{non}({\mathcal N}), {\mathfrak d}\) and \({\mathfrak s}_d\geq{\mathfrak b}\). Also, the following consistency results are proved: \({\mathfrak r}_d>{\mathfrak b}\), \({\mathfrak s}_d<{\mathfrak d}\), \({\mathfrak s}_d<{\mathfrak r}\), \({\mathfrak r}_d<\text{add}({\mathcal M})\) and \({\mathfrak s}_d>\text{cof}({\mathcal M})\). To prove the consistency of the two latter inequalities, the author introduces new cardinal invariants \({\mathfrak r}_{\text{pair}}\) and \({\mathfrak s}_{\text{pair}}\). Some relations between \({\mathfrak r}_{\text{pair}}\), \({\mathfrak s}_{\text{pair}}\) and other cardinal invariants are proved as well.
scientific article; zbMATH DE number 5708672
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