the sublattice induced on $\{b,d,e,f,h\}$ is as shown in the figure. It is well known that no lattice with this lattice as a sublattice is distributive.

Downset Construction {#sec:downsets}
====================

Our main result of this section is Theorem \[thm:char2\]. Before we prove it, we make some easy observations about the construction $G(P,A)$ of Definition \[def:GPA\]. Recall that $A$ is a sub-digraph of a poset $P$; it will always have the same vertex set as $P$ We write $a \to b$ to mean that $(a,b)$ is an arc of $A$.

An arc $(x', y')$ of $P$ is [*useless*]{} in a sub-digraph $A$ if there is $(x,y) \not\in A$ with $x' \leq x \leq y \leq y'$ (and $x' \neq x$ or $y' \neq y$). Removing all useless arcs from $A$ it clearly satisfies the following directed version of $$\label{id:minmaxd}
     (u' \leq u \leq v \leq v' \mbox{ and } u' \to v')  \Rightarrow u \to v$$

\[lem:extension1\] The graph $G(P,A)$ is unchanged by adding or removing useless arcs from $A$. Thus $A$ may be assumed to satisfy .

Let $x' \leq x \leq y \leq y'$, and $x \not\to y$ in $A$. For any two downsets $D$ and $D'$ of $P$ 