Proposition 3.10

Let $L$ be a Boolean algebra such that every sequence $(a_n)_{n\in\mathbb N}$ in $L$ has a least upper bound, and let $B\subseteq L$ be an inclusion-maximal nonempty subset excluding $\bot$ in which any two members have a common lower bound belonging to $B$. Then the following are equivalent: $B$ is nonempty, excludes $\bot$, is upward closed and closed under binary meets, and for every sequence $(a_n)_{n\in\mathbb N}$ and every least upper bound $b$ of its range, $b\in B$ implies $a_n\in B$ for some $n$; and for every sequence $(a_n)_{n\in\mathbb N}$ whose terms all belong to $B$, every greatest lower bound $b\in L$ of its range belongs to $B$. The quantified sequences include constant sequences and sequences with repeated terms; the statement uses sequences indexed by all natural numbers, not empty families.