Is there a characterization of hyperplane arrangement intersection posets?

For a hyperplane arrangement $\mathcal{A}$ over a vector space $V$, we define its intersection poset, $L(\mathcal{A})$, as the set of all nonempty intersections of hyperplanes in $\mathcal{A}$ ordered by reverse inclusion. The empty intersection, $V$ itself, is the unique minimal element of $L(\mathcal{A})$.

It is known that $L(\mathcal{A})$ is a ranked meet-semilattice, and moreover, any interval $[x,y]$ in $L(\mathcal{A})$ is a geometric lattice. But these properties alone are not sufficient for some poset $P$ to be the intersection poset of a hyperplane arrangement. Consider the following poset:poset http://www.freeimagehosting.net/newuploads/b4quk.png

If this were the intersection poset of some arrangement, then $a$ would be parallel to $d$ and to $c$, $b$ would be parallel to $d$, and thus $b$ and $c$ would be parallel. But $b$ and $c$ have nonempty intersection, so this is nonsense.

Is there a known characterization of hyperplane arrangement intersection posets?

– jeq
Aug 8 '17 at 0:52

The intersection poset of a (not necessarily central) hyperplane arrangement is a geometric semi-lattice, as defined by Bjorner and Wachs, who show that every such poset is isomorphic to the subposet of $x \not \geq a$ of a geometric lattice and an atom a. This corresponds geometrically to putting the arrangement into projective space, and adding the hyperplane at infinity, which corresponds to $a$. I haven't looked at the definition, but I suspect your example above is not a geometric semi-lattice; if it were it should be "realizable" by an arrangement of pseudo-hyperplanes, but your argument shows it cannot.
Given a geometric semi-lattice, it is realizable by an arrangement of affine hyperplanes if and only if the associated geometric lattice is realizable by linear hyperplanes, which is equivalent to the matroid-realizability question as Richard points out above. For this you need to specify a field - some lattices are realizable over $\mathbb C$ or over a finite field, but not over $\mathbb R$. Oriented matroids and arrangements of pseudo-spheres are relevant only to realizability over $\mathbb R$.