Maximal set of pairwise disjoint elements of a dense subset.

Question

Let $B$ be a complete Boolean algebra, and suppose $D \subseteq B$ is a dense subset. How is it possible to construct a maximal set of pairwise disjoint elements of $D$? Is it true that $\sum S = \sum D$ ? I want to do this since it is used frequently to construct partitions of $B$. Any help would be appreciated.

Answer 1

You know that $\bigvee D=1$. If $\bigvee S=s<1$, there is a $d\le\lnot s$ in $D$, and $S\cup\{d\}$ is a pairwise disjoint subset of $D$ strictly larger than $S$.

To get a maximal pairwise disjoint subset of $D$, I'd simply appeal to the axiom of choice.