an ordering of its vertices such that if it satisfies the so-called [*min-max*]{} identity: $$\label{id:minmax}
   (u' \leq u \leq v \leq v' \mbox{ and } u' \sim v')  \Rightarrow u \sim v$$ We take this as our definition of a proper interval graph in the reflexive context, and say a proper interval graph is in [*min-max form*]{} if its vertices are labelled $\{0,1, \dots, n\}$ for some $n$ so that it satisfies .

Simple arguments (see Fact \[fact:ids\]) show that any reflexive graph that is compatible with a chain lattice is a proper interval graph. As any distributive lattice is embeddable in a product of chains, it follows that any $\operatorname{DL}$-graph is a subgraph of a categorical product (defined in Section \[sec:def\]) of proper interval graphs. In fact Dilworth [@Di50] showed, and we recall this in more detail in Section \[sec:main2\], that any chain decomposition of $P$ yields an embedding of $\cD(P)$ into a product of chains. The embedding shown in Figure \[fig:GPA\] comes from the decomposition of $P$ into the chains $a\prec c$ and $b \prec d$. Figure \[fig:PI\] shows the embedding corresponding to the decomposition of $P$ into the chains $a$, $b \prec c$, and $d$. For the 