$x_1, \dots, x_l$ we may express each hyperplane $H \in {{\mathcal A}}$ as the zero set of a degree $1$ polynomial $\alpha_H \in {\mathbb{C}}[x_1, \dots, x_l]$. The product $Q({{\mathcal A}})=\Pi_{H \in {{\mathcal A}}}\alpha_H$ is called the defining polynomial of ${{\mathcal A}}$. The [*complement*]{} of ${{\mathcal A}}$ is $M_{{{\mathcal A}}} := {\mathbb{C}}^l \setminus Q({{\mathcal A}})^{-1}(0)$.

For arrangements with the property that $\bigcap_{H \in {{\mathcal A}}}H \neq \emptyset$, by a choice of coordinates, one can assume that all hyperplanes contain the origin, hence all corresponding equations $\alpha_H$ are homogeneous. Such an arrangement is called [*central*]{}.

Recall that the [*intersection lattice*]{} $\mathcal L({{\mathcal A}})$, i.e., the set of all subspaces which are intersections of hyperplanes of ${{\mathcal A}}$, is a poset with a rank function, whose partial order is given by reverse inclusion, and rank given by $r(\cap_{H\in {{\mathcal B}}} H) := \operatorname{codim}(\cap_{H\in {{\mathcal B}}} H)$. The minimal element of $\mathcal L({{\mathcal A}})$ is $V$, seen as intersection of the empty set of hyperplanes.

\[latice\] The lattice of the [*boolean*]{} arrangement ${{\mathcal A}}\subset {\mathbb{C}}^l,\; Q({{\mathcal A}})=\Pi_{i=1}^l x_i$, contains $2^l$ subspaces $H_{i_1} \cap \dots \cap H_{i_s}$, indexed by all ordered subsets $i_1 < i_2< \dots <i_s$ of $\{1, 2, \dots, l\}$, with rank $s$. 