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Reaping Numbers of Boolean Algebras
Author(s) -
Dow Alan,
Steprāns Juris,
Watson Stephen
Publication year - 1996
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/28.6.591
Subject(s) - mathematics , free boolean algebra , complete boolean algebra , cardinality (data modeling) , boolean algebras canonically defined , two element boolean algebra , stone's representation theorem for boolean algebras , partition (number theory) , boolean algebra , combinatorics , discrete mathematics , algebra over a field , pure mathematics , filtered algebra , computer science , data mining
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 r n , 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 r n +1,2 (B) ≠ r n ,2 (B). Also, { r n , m (B): m ⩽ n ∈ ω} consists of at most two consecutive cardinals. The existence of a Boolean algebra B such that r n , m (B) ≠ r n ′, m ′ (B) is equivalent to a statement in finite combinatorics which is also discussed.