Almost refinement, reaping, and ultrafilter numbers

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 $\aleph_1$.

💡 Research Summary

The paper studies the combinatorial structure of maximal antichains in a c.c.c. Boolean algebra B when equipped with the “almost refinement” order ≤. Two antichains A and B satisfy A ≤ B if, after removing finitely many elements from A and taking the join of those removed elements, the resulting antichain refines B. The authors view the collection Part*(B)=⟨Part(B),≤,Part(B)⟩ as a relational system and analyse its cardinal invariants – the dominating number d(Part(B)) and the bounding number b(Part*(B)) – via generalized Galois‑Tukey connections.

Section 2 recalls the framework of relational systems ⟨A⁻,A,A⁺⟩, defines d(A) and b(A), and introduces generalized Galois‑Tukey connections ⟨φ⁻,φ⁺⟩: A ≤_T B. The basic facts that A ≤_T B implies d(A) ≤ d(B) and b(B) ≤ b(A) are recorded, together with operations such as sequential composition (A;B) and the σ‑extension A^σ, which will be used later.

In Section 3 the almost‑refinement relation is defined for maximal antichains of a c.c.c. Boolean algebra. Lemma 3.4 shows that for any a∈B⁺ the system Part*(B↾a) Galois‑Tukey reduces to Part*(B). Proposition 3.7 characterises when the dominating and bounding numbers are uncountable: d(Part*(B))>ℵ₀ iff B is non‑atomic, while b(Part*(B))>ℵ₀ iff B is weakly ⟨ω,ω⟩‑distributive (equivalently, every countable family of maximal antichains can be met by a single antichain only finitely often). Theorem 3.8 establishes the chain ⟨B⁺,≥,B⁺⟩^σ ≤_T Part*(B) ≤_T Part(B), hence Part*(B) and Part(B) share the same dominating number (Corollary 3.9).

Theorem 3.11 treats non‑atomic σ‑finite c.c. algebras. Using a decomposition B⁺=⋃ₙSₙ where each Sₙ contains only finite antichains, the authors construct maps φ⁻:ω^ω→Part(B) and φ⁺:Part(B)→ω^ω satisfying φ⁻(f) ≤* B ⇒ f ≤* φ⁺(B). Consequently ⟨ω^ω,≤,ω^ω⟩ ≤_T Part(B), showing that the almost‑refinement structure is at least as complex as the classical eventual domination order.

The paper then focuses on the Cohen algebra C_ω = B(2^ω)/M, where M is the meagre ideal. Lemma 3.13 provides a canonical representation of maximal antichains by basic clopen sets N_s, and Theorem 3.14 proves a Galois‑Tukey equivalence ⟨nwd(2^ω),⊆,nwd(2^ω)⟩ ≡_T Part*(C_ω) ≡_T Part(C_ω). Thus the almost‑refinement order on C_ω mirrors the inclusion order on nowhere‑dense subsets of the Cantor space.

Section 4 introduces the reaping relation for a Boolean algebra B and its associated cardinal invariants (the reaping and splitting numbers). The authors show that the reaping relation of the reduced power B^ω/fin is linked both to Part*(B) and to the classical reaping relation on P(ω)/fin. In particular, they completely determine the reaping and splitting numbers of the reduced power of the Cohen algebra, obtaining that both are equal to cof(ℳ), the cofinality of the meagre ideal.

Finally, Section 5 applies these results to the ultrafilter number u(B). Using the earlier Galois‑Tukey connections, they prove cof(ℳ) ≤ u(C_ω), parallel to Burk’s result cof(ℕ) ≤ u(B_ω) for the random algebra. Moreover, by extending the parametrized diamond principles of Moore, Hrušák, and Džamonja to a class of “Borel‑homogeneous” Boolean algebras, they show that if the diamond principle associated with the reaping relation of the reduced power of C_ω holds, then u(C_ω)=ℵ₁. This provides a consistency result: under suitable combinatorial hypotheses the ultrafilter number of the Cohen algebra can be forced down to the minimal possible uncountable cardinal.

Overall, the paper unifies several cardinal characteristics—dominating, bounding, reaping, splitting, and ultrafilter numbers—through the lens of almost refinement and Galois‑Tukey theory, yielding new insights especially for the Cohen algebra and its reduced powers.