Let us say that a complete Boolean algebra $B$ is:

**countably distributive**when for any sequence $(I_n)_{n\in\mathbb{N}}$ of sets and any elements $(u_{n,i})_{n\in\mathbb{N},i\in I_n}$ of $B$ we have $$\bigwedge _ {n\in\mathbb{N}}\, \bigvee _ {i\in I_n} u_{n,i} = \bigvee _ {f\in\prod_{n\in\mathbb{N}} I_n} \, \bigwedge_{n\in\mathbb{N}} u_{n,f(n)}$$**countably compact**when for any sequence $(v_n)_{n\in\mathbb{N}}$ of elements of $B$, if $\bigwedge_{n\in\mathbb{N}} v_n = 0$ then there is $F \subseteq \mathbb{N}$ finite such that $\bigwedge_{n\in F} v_n = 0$.

**Questions:** Is here a relation between these two properties (does one imply the other)? If not, might I have examples of Boolean algebras with one but not the other?

(The apparent lack of any mention of the second property everywhere I searched in the literature leaves me to suspect that there is something deeply wrong with this property which I failed to see, so my question is perhaps stupid: if so, I apologize. But, generally speaking, I am interested in what can be said about this “countably compact” property.)