are downsets, this reduces to the statement that for all $(y, x)$ in ${{A^{\rm c}}}$ with $x \in D$, $y \not\in D'$.

\[thm:char2\] For any reflexive graph $G$ compatible with a distributive lattice $L$, $G {\buildrel {} \over \cong}G(J_L, A)$ for a unique sub-digraph $A$ of $J_L$ satisfying .

Let $G$ be compatible with a distributive lattice $L$. By [@Bi] we have that $L {\buildrel {} \over \cong}\cD(J_L)$, so we denote vertices of $G$ by downsets of the poset $J_L$.

We define a sub-digraph $A = A(G,J_L)$ of $J_L$ as follows. For a vertex $p$ of $J_L$, let $$C_p = X {-}{[ p \bm{\rangle}} = \bigcup \{ X \in \cD \mid p \not\in X \}$$ be the maximum downset not containing $p$. For each arc $y \to x$ of $J_L$, let $y \to x$ be in $A$ if ${\bm{\langle} x ]} \cap C_y$ and ${\bm{\langle} x ]}$ are adjacent in $G$. We show that $G {\buildrel {} \over \cong}G(J_L, A)$.

Let $D$ and $D'$ be adjacent downsets of $J_L$. To show that $D$ and $D'$ are adjacent in $G(J_L,A)$, it is enough to show, without loss of generality, that any arc $(y,x) \in J_L$ for $x,y \in (D {-}D')$, is 