Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 1025–1036

# Cofinality of the laver ideal

• Miroslav Repický
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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