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 $\{b\in P:b\wedge a\ne \varnothing \}$ 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 $r_{n+1,2}(B)\ne r_{n,2}(B)$. 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 $r_{n,m}(B)\ne r_{n^{\prime },m^{\prime }}(B)$ is equivalent to a statement in finite combinatorics which is also discussed.