Proposition 3.5

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 $B\subseteq L$ excluding $\bot$ such that any two members have a common lower bound in $B$. Put $D(a)=\{B\in\mathcal Q(L):a\in B\}$ and give $\mathcal Q(L)$ the topology generated by these sets. Say that $L$ is distributive when both $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$ and $a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c)$ hold for every $a,b,c$. Then the theorem asserts three equivalences: $L$ is distributive if and only if $D(a)\cup D(b)=D(a\vee b)$ for every $a,b$; $L$ is distributive if and only if $D(a)\cup D(a^\perp)=\mathcal Q(L)$ for every $a$; and $L$ is distributive if and only if every subset of $\mathcal Q(L)$ that is both open and closed equals $D(a)$ for some $a\in L$.