boolean arrangement are modular.

\[stanley\] A central arrangement ${{\mathcal A}}\subset \mathbb{C}^{l}=V$ is called [*supersolvable*]{} if there is a maximal chain of modular elements: $$\label{eq:21}
V=X_{0}\supset X_{1}\supset\dots \supset X_{r}=C({{\mathcal A}})$$ where $C({{\mathcal A}})= \bigcap_{H \in {{\mathcal A}}}H$ is the center.

1.  \[bool3\] The boolean arrangement from Example \[latice\] is supersolvable, with a maximal chain of modular elements given for instance by $V \supset \{x_1=0\}\supset \{x_1=x_2=0\} \dots \supset \{x_1= \dots=x_l=0\}$.

2.  \[braidchain\] A chain of modular elements of length $l-1$ for the braid arrangement in Example \[essential\] (\[converse\]) is the following: $V \supset \{x_1=x_2\} \dots \supset \{x_1= \dots=x_l\}$.

The next definition gives a topological interpretation of supersolvability: as shown by Terao [@Te], $\mathcal L({{\mathcal A}})$ is supersolvable if and only if ${{\mathcal A}}$ is fiber-type.

\[fibertype\] An essential arrangement ${{\mathcal A}}\subset \mathbb{C}^{l}$ is said to be [*fiber-type*]{} if there is a sequence of subarrangements in ${{\mathcal A}}$ of length $l$ : $$\label{eq:22}
\{ H \}={{\mathcal A}}_{1} \subset {{\mathcal A}}_{2} \subset \dots \subset {{\mathcal A}}_{l}={{\mathcal A}}$$ so that there is a tower of topological fibrations $M_{{{\mathcal A}}_{i}}\longrightarrow M_{{{\mathcal A}}_{i-1}}$, induced by linear maps, with fiber $F_i$ a complex line with a finite number of points removed.

Fiber-type arrangements are a class of arrangements 