supersolvable.

Hypersolvable arrangements {#sect:special}
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Jambu-Papadima [@JP1] introduced hypersolvable arrangements. This class generalizes the fiber-type class, preserving many of its topological and combinatorial properties.

Inspired by the combinatorics of the braid arrangement, the conditions in the inductive definition of hypersolvability involve only collinearity relations, or equivalently elements in the lattice of the arrangement up to rank $2$.

The building blocks are certain pairs of central arrangements, $({{\mathcal A}}, {{\mathcal B}}),\; {{\mathcal B}}\subset_{\neq} {{\mathcal A}}$, with the convention that elements in ${{\mathcal B}}$ are denoted by $\alpha, \beta, \gamma, \dots$ and elements in $\overline{{{\mathcal B}}}:={{\mathcal A}}\setminus{{\mathcal B}}$ by $a, b, c, \dots$. We will identify the hyperplanes in ${{\mathcal A}}\subset V$ with points in $\mathbb{P}(V^*)$.

\[exthyper\] An arrangement pair $({{\mathcal A}}, {{\mathcal B}})$ is a [*solvable extension*]{} if the following conditions are satisfied:

1.  **closure**: any distinct points $\alpha, \beta, a$, with $\alpha, \beta \in \mathcal{B}$ and $a \in \overline{\mathcal{B}}$ are non-collinear;

2.  **completion** : for any two distinct points $a, b \in \overline{\mathcal{B}}$, there is a point $\alpha \in {\mathcal{B}}$ on the line determined by $a$ and $b \,$; if the closure condition holds, $\alpha=:f(a,b)$ is uniquely determined;

3.  **solvability**: for any distinct $a,b,c$ in $\overline{\mathcal{B}}$, the three points $f(a,b),f(c,b)$, 