WEAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY

Let B be a complete Boolean algebra. We show that if lambda is an infinite cardinal and B is weakly (lambda(omega), omega)-distributive, then B is (lambda, 2)-distributive. Using a similar argument, we show that if kappa is a weakly compact cardinal such that B is weakly (2(kappa), kappa)-distributive and B is (alpha, 2)-distributive for each alpha < kappa, then B is (kappa, 2)-distributive.