Proposition 3.6

Let $L$ be a bounded lattice with a function $a\mapsto a^\perp$ satisfying $a\wedge a^\perp=\bot$, $a\vee a^\perp=\top$, and $(a^\perp)^\perp=a$ for every $a$; $(a\wedge b)^\perp=a^\perp\vee b^\perp$ and $(a\vee b)^\perp=a^\perp\wedge b^\perp$ for every $a,b$; and $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$ whenever $b\le a$ and $c\le a^\perp$. Let $\mathcal Q(L)$ consist of the inclusion-maximal nonempty subsets of $L$ excluding $\bot$ such that any two members have a common lower bound in the subset, and equip it with the topology generated by $D(a)=\{C\in\mathcal Q(L):a\in C\}$. For every $B\in\mathcal Q(L)$, the singleton $\{B\}$ is open if and only if there exists exactly one $a\in L$ such that $a$ is an atom, meaning $a\ne\bot$ and every $x\le a$ is either $\bot$ or $a$, and $B=\{b\in L:a\le b\}$.