Lattice and Order Properties of the Poset of Regions in a Hyperplane Arrangement

We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a supersolvable arrangement of rank k − 1. An explicit description of the doublings leads to a proof that the order dimension of the poset of regions (again with respect to a canonical base region) of a supersolvable hyperplane arrangement is equal to the rank of the arrangement. In particular, the order dimension of the weak order on a finite Coxeter group of type A or B is equal to the number of generators. The result for type A (the permutation lattice) was proven previously by Flath [11]. We show that the poset of regions of a simplicial arrangement is a semi-distributive lattice, using the previously known result [2] that it is a lattice. A lattice is congruence uniform (or “bounded” in the sense of McKenzie [18]) if and only if it is semi-distributive and congruence normal [7]. Caspard, Le Conte de Poly-Barbut and Morvan [4] showed that the weak order on a finite Coxeter group is congruence uniform. Inspired by the methods of [4], we characterize congruence normality of a lattice in terms of edge-labelings. This leads to a simple criterion to determine whether or not a given simplicial arrangement has a congruence uniform lattice of regions. In the case when the criterion is satisfied, we explicitly characterize the congruence lattice of the lattice of regions.