Lemma 3.1

Let $L$ be a bounded lattice and let $a\mapsto a^\perp$ be a function $L\to L$ satisfying the following identities: for every $x$, $x\wedge x^\perp=\bot$, $x\vee x^\perp=\top$, and $(x^\perp)^\perp=x$; for every $x,y$, $(x\wedge y)^\perp=x^\perp\vee y^\perp$ and $(x\vee y)^\perp=x^\perp\wedge y^\perp$; and for every $x,y,z$ with $y\le x$ and $z\le x^\perp$, $x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)$. 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 $B$, and set $D(x)=\{B\in\mathcal Q(L):x\in B\}$. For any $a,b\in L$, if $D(a)=D(b)$, then $a=b$.