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References (14)
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(1991)
Reaping numbers, Handwritten notes -
There are Boolean algebras B 1 , B 2 and B 3 such that • r 9,5 (B) ≤ r 2,2 (B) -
Proof: Let n = 3 -
Amer Bešlagić, E. Douwen (1990)
Spaces of nonuniform ultrafilters in spaces of uniform ultrafiltersTopology and its Applications, 35
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Corollary 4.2. For any integer k ≥ 2 there is some Boolean algebra B such that r 2k+1,3 (B) ≤ r k -
Dan VELLEMAN1 (1984)
co-MORASSES, AND A WEAK FORM OF MARTIN'S AXIOM PROVABLE IN ZFC -
3 (B) ≤ r 4,2 (B). Proof: Use -
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Question 4.4. Does (11, 6) ≺ (3, 2) hold? References -
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m) ≺ (i, j) if and only if r n,m (B) ≤ r i,j (B) for every Boolean algebra B. Obviously ≺ is a transitive relation and so it induces a partial order on equivalence classes
- Publisher
- Wiley
- Copyright
- © London Mathematical Society
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/blms/28.6.591
- Publisher site
- See Article on Publisher Site
Abstract
A subset A of a Boolean algebra B is said to be (n,m)‐reaped if there is a partition of unity p ⊂ B of size n such that |{b ∈ p:b ∧ a ≠ 0}| ⩾ m for all a ∈ A. The reaping number rn,m (B) of a Boolean algebra B is the minimum cardinality of a set A ⊂ B∖{0} which cannot be (n,m)‐reaped. It is shown that for each n∈ω, there is a Boolean algebra B such that rn+1,2(B) ≠ rn,2(B). Also, {rn,m(B):m⩽n ∈ ω} consists of at most two consecutive cardinals. The existence of a Boolean algebra B such that rn,m (B) ≠ rn′,m′ (B) is equivalent to a statement in finite combinatorics which is also discussed.
Journal
Bulletin of the London Mathematical Society – Wiley
Published: Nov 1, 1996