(see for example [@OM Definition 4.1.1]), which is the oriented matroid corresponding to the hyperplane arrangement. In fact, some of our results extend to general oriented matroids. In this article we have chosen to concentrate on hyperplane arrangements to keep the exposition more concrete. In particular, we do not assume that the reader knows what an oriented matroid is.

We denote by ${\mathscr{L}}$ the set of nonempty subsets of $V$ of the form $C=s^{-1}(X)$, for a sign vector $X\in\{+,-,0\}^E$. The elements of ${\mathscr{L}}$ are called *faces* of the arrangement. The set ${\mathscr{L}}$ has a natural partial order given by $C \leq D$ if and only if $C \subseteq \overline{D}$. The relation $C \leq D$ is equivalent to the fact that for every $e\in E$ we have $s(C)_e=0$ or $s(C)_e=s(D)_e$. The set ${\mathscr{L}}$ with this partial order is called the *face poset* of the arrangement. Note that $V_{0}$ is the minimal element of ${\mathscr{L}}$. When we adjoin a maximal element $\widehat{1}$ to the poset ${\mathscr{L}}$, we obtain a lattice ${\mathscr{L}}\cup \{\widehat{1}\}$ known as the *face lattice*. Note that, under our convention, faces other than $V_0$ are not closed subsets of $V$: for every $C\in{\mathscr{L}}$, the closure $\overline{C}$ is a closed convex 