called a [*defining polynomial*]{} of ${\mathcal{A}}$.

If ${\mathcal{B}}$ is a subset of ${\mathcal{A}}$, it is called a [*subarrangement*]{} of ${\mathcal{A}}$. The [*intersection semilattice*]{} of ${\mathcal{A}}$, denoted by $L({\mathcal{A}})$, is the set of all non-empty intersections of elements of ${\mathcal{A}}$, i.e. $$L({\mathcal{A}})=\left\{\textstyle{\bigcap_{H \in {\mathcal{B}}} H} \mid {\mathcal{B}} \subseteq {\mathcal{A}} \right\}.$$ These subspaces are called the [*flats*]{} of ${\mathcal{A}}$. The set $L({\mathcal{A}})$ is partially ordered by reverse inclusion.

The [*complement*]{} of ${\mathcal{A}}$ is defined as $$M({\mathcal{A}}) = V \setminus \bigcup_{H \in {\mathcal{A}}} H .$$ The complement of an arrangement ${\mathcal{A}}$ in ${\mathbb{R}}^n$ is clearly disconnected. It is made up of separate pieces called [*chambers*]{} or [*regions*]{}, each of which may be either bounded or unbounded.

Each flat of ${\mathcal{A}}$ is also divided into sections by the hyperplanes that do not contain the flat; these sections are called the [*faces*]{} of ${\mathcal{A}}$. Chambers are faces because the whole space is a flat. The faces of codimension $1$ may be called the [*facets*]{} of ${\mathcal{A}}$. The [*face semilattice*]{} of an arrangement is the set of all faces, ordered by inclusion. The arrangement ${\mathcal{A}}$ said to be [*essential*]{} if the minimal dimensional flats are points (that we call *vertices* of the arrangement).

Every arrangement ${\mathcal{A}}_{\mathbb{R}}$ 