Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 1025–1036 | Cite as

Cofinality of the laver ideal

Article

Abstract

Yurii Khomskii observed that \({{\mathrm{cof}}}(l^0)>\mathfrak {c}\) assuming \(\mathfrak {b}=\mathfrak {c}\) and he asked whether the inequality \({{\mathrm{cof}}}(l^0)>\mathfrak {c}\) is provable in ZFC. We find several conditions that imply some variants of this inequality for tree ideals. Applying a recent result of Brendle, Khomskii, and Wohofsky we show that \(l^0\) satisfies some of these conditions and consequently, \({{\mathrm{cof}}}(l^0)=\mathfrak {d}(^\mathfrak {c}l^0)\ge \mathfrak {d}(^\mathfrak {c}\mathfrak {c})>\mathfrak {c}\). We also prove that if the cellularity of a Boolean algebra B is hereditarily \(\ge \kappa \), then every \(\kappa \)-sequence in \(B^+\) has a \(\kappa \)-subsequence with a disjoint refinement.

Keywords

Laver ideal Cardinal invariants Disjoint refinements 

Mathematics Subject Classification

Primary 03E17 Secondary 03E15 03E35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovak Republic

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