Reaping Numbers of Boolean Algebras

Abstract

A subset $A$ of a Boolean algebra $B$ is said to be $(n,m)$-reaped if there is a partition of unity $P \subset B$ of size $n$ such that the cardinality of $\{b \in P: b \wedge a \neq \emptyset\}$ is greater than or equal to $m$ for all $a\in A$. The reaping number $r_{n,m}(B)$ of a Boolean algebra $B$ is the minimum cardinality of a set $A \subset B\setminus \{0\}$ such which cannot be $(n,m)$-reaped. It is shown that, for each $n \in \omega$, there is a Boolean algebra $B$ such that $r_{n+1,2}(B) \neq r_{n,2}(B)$. Also, $\{r_{n,m}(B) : \{n,m\}\subseteq\omega\}$ consists of at most two consecutive integers. The existence of a Boolean algebra $B$ such that $r_{n,m}(B) \neq r_{n',m'}(B)$ is equivalent to a statement in finite combinatorics which is also discussed.