Almost refinement, reaping, and ultrafilter numbers

TL;DR

This paper explores the combinatorial properties of maximal antichains and reaping relations in Boolean algebras, establishing new bounds and equalities for ultrafilter numbers in relation to cardinal invariants and set-theoretic principles.

Contribution

It introduces new results connecting ultrafilter numbers of Cohen algebras with cofinalities of ideals and set-theoretic principles, advancing understanding of Boolean algebra invariants.

Findings

Ultrafilter number of Cohen algebra ≥ cofinality of meagre ideal

A parametrized diamond principle can make ultrafilter number of Cohen algebra equal to 

New insights into the structure of maximal antichains in Boolean algebras

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 $\aleph_1$.