polyhedral cone in $V$, and it is an intersection of closed half-spaces $\overline{H^{\pm}_{\alpha_e}}$. The poset ${\mathscr{L}}$ is graded with the rank of a face $C\in{\mathscr{L}}$ given by $\rho(C) = \dim(C)-\dim(V_0)$, where we write $\dim(C)$ for $\dim(\operatorname{Span}(C))$.

We denote by ${\mathscr{T}}$ the set of maximal faces of ${\mathscr{L}}$. These elements are often called *chambers*, *regions* or *topes*, and are the connected components of $V-\bigcup_{e\in E}H_{\alpha_e}$. If $T \in {\mathscr{T}}$ then $T$ is an open subset of $V$, and its closure is a closed convex polyhedral cone of dimension $\dim(V)$.

If $X,Y\in\{+,-,0\}^E$, their *composition* $X\circ Y$ is the sign vector defined by $$(X\circ Y)_e = \begin{cases}X_e & \text{ if }X_e\not=0,\\
Y_e & \text{ otherwise.}\end{cases}$$ If $C,D\in{\mathscr{L}}$ then $s(C)\circ s(D)$ is also the image of a face of ${\mathscr{L}}$, and we denote this face by $C\circ D$. This is the unique face of ${\mathscr{L}}$ that contains all vectors of $V$ of the form $x+\varepsilon y$, with $x\in C$, $y\in D$ and $\varepsilon>0$ sufficiently small (relative to $x$ and $y$). Define the *separation set* of $C$ and $D$ to be the set $$S(C,D)=\{e \in E :  s(C)_e=-s(D)_e\not=0\}.$$ This is the set of $e \in E$ such that $C$ and $D$ are on different sides 