Almost refinement, reaping, and ultrafilter numbers
Jörg Brendle, Michael Hrušák, Francesco Parente
TL;DR
The paper extends Matet's almost refinement from partitions of $\omega$ to maximal antichains in cc Boolean algebras, using generalized Galois-Tukey connections to relate almost refinement to classical relational systems and cardinal invariants. It identifies a tight link between $\mathop{\mathrm{Part}}^*(\mathbb{B})$ and the nowhere dense ideal for the Cohen algebra, and analyzes the reaping relation to determine exact reaping/splitting numbers for reduced powers, including $\prescript{\omega}{}{\mathbb{C}_\omega}/\mathrm{Fin}$. The ultrafilter number for reduced powers is shown to be controlled by the base algebra, with $\cof(\mathcal{M})$ bounding $\mathfrak{u}(\mathbb{C}_\omega)$ and, under a parametrized diamond principle, $\mathfrak{u}(\mathbb{C}_\omega)=\aleph_1$, linking these invariants to Borel-homogeneous forcings and Cichoń’s diagram. Overall, the work connects combinatorial properties of antichains in Boolean algebras with key cardinal invariants via GT-connections and diamond principles.
Abstract
We investigate the combinatorial structure of the set of maximal antichains in a Boolean algebra ordered by almost refinement. We also consider the reaping relation and its associated cardinal invariants, focusing in particular on reduced powers of Boolean algebras. As an application, we obtain that, on the one hand, the ultrafilter number of the Cohen algebra is greater than or equal to the cofinality of the meagre ideal and, on the other hand, a suitable parametrized diamond principle implies that the ultrafilter number of the Cohen algebra is equal to $\aleph_1$.