embedding in Figure \[fig:PI\] the graph $G(P,A)$ is an induced subgraph of proper interval graphs, in this case paths, on the chain factors. It turns out that this happens when certain edges in the complement of $A$ in $P$ are contained in the chain decomposition of $P$.

For any $\operatorname{DL}$-pair $(G,L)$, we get, in Theorem \[thm:embedding\], an embedding of $L$ into a product of chains such that $G$ is an induced subgraph of $\cG$. Further, it is an induced subgraph of quite a particular form.

The vertex set of a product $\cG = \prod_{i = 1}^d G_i$ of proper interval graphs $G_i$ is a set of $d$-tuples $x = (x_1, \dots, x_d) \in \prod_{i = 1}^d \{0,1, \dots, n_i\}$ for some $n_1, \dots, n_d$. A [*vertex interval*]{} of $\cG$ is the set $${{[{{\alpha}_{[i]}}, {{\beta}^{[j]}}]}}= \{ x \in V(\cG) \mid  \alpha \leq x_i \mbox{ and } x_j \leq \beta \}$$ for some $i,j \in [d]$, $\alpha \leq n_i$ and $\beta \leq n_j$. (See right side of Figure \[fig:PI\].)

(14,5)(0,0) (6,0)[![Left: The lattice $\cD(P)$ from Figure \[fig:GPA\] embedded in a product of three chains, and the graph $G(P,A)$ from Figure \[fig:GPA\] embedded as an induced subgraph of the product of paths on 