Reaping numbers and operations on Boolean algebras (Q1966139)

For \(n\in N\setminus \{1\}\), an \(n\)-partition of a Boolean algebra \(A\) is a disjoint system \( \langle a_i; i<n \rangle \) of elements of \(A\) s.t.\ \(\sum _{i<n} a_i = 1\). The set \(X\subseteq A\) is said to be \(n\)-reaped by an \(n\)-partition \( \langle a_i; i<n \rangle \) if for all \(x\in X^+\) the set \(\{i<n; \;a_i \cdot x = 0\}\) has size greater than 1. For \(n\in N\setminus \{1\}\), let \(r_n (A) = \min \{|X |\); \(X\subseteq A\) and \(X\) can not be \(n\)-reaped by any \(n\)-partition\(\}\) and \(r_{\omega }(A) = \sup \{r_n(A); n\in N\setminus \{1\}\}\). Some results concerning the numbers \(r_n(A)\) and \(r_{\omega } (A)\) are proved.