\subsection{Order-restriction DAGs}\label{sec:order-dags}

An \emph{order-restriction DAG} $D$ consists of a directed acyclic graph
on a finite set $V$, together with an open interval
$(\ell_x,r_x)\subseteq(0,1)$ for each vertex $x$. Its \emph{feasible
assignments} are the points in
\begin{equation}\label{eq:node-region}
K(D)=\{X\in\R^V:\ell_x<X_x<r_x\text{ for every }x,
\quad X_x<X_y\text{ for every arc }x\to y\}.
\end{equation}
Assignments with distinct coordinates determine relative orders of the
vertices. For example, the arcs $a\to b$ and $a\to c$, with all three
intervals equal to $(0,1)$, allow both $a<b<c$ and $a<c<b$.
Write $V(D)=\vol K(D)$, taking $V(D)=1$ when $V$ is empty.

We call the intervals \emph{canonical} when each is the full range of
its coordinate in feasible assignments.