Given a complete boolean algebra B, and two partitions W and T of B, why is it true that W induces a partition on every element of T? (And is this true more generally - does W induce a partition on every non zero element of B?)
This is motivated by trying to understand the solution given here : If $B$ is an infinite complete Boolean algebra, then its saturation is a regular uncountable cardinal to theorem 7.15 in Jech's Set Theory.