  The shard intersection order is a new lattice structure on a finite Coxeter
group W which encodes the geometry of the reflection arrangement and the
lattice theory of the weak order. In the case where W is the symmetric group,
we characterize shard intersections as certain pre-orders which we call
permutation pre-orders. We use this combinatorial characterization to determine
properties of the shard intersection order. In particular, we give an
EL-labeling.
