Biclosed sets in real hyperplane arrangements

TL;DR

This paper explores the structure of chambers in real hyperplane arrangements, establishing a correspondence with biclosed sets in cases where the chambers form a lattice, thus generalizing previous results.

Contribution

It introduces the concept of biclosed sets as a generalization of biconvex sets for chambers in arrangements with lattice structures.

Findings

Chambers in certain arrangements correspond to biclosed sets.

The correspondence extends known results from simplicial and supersolvable arrangements.

Biclosed sets provide a weaker condition than biconvexity, broadening the class of arrangements studied.

Abstract

The set of chambers of a real hyperplane arrangement may be ordered by separation from some fixed chamber. When this poset is a lattice, Bjorner, Edelman, and Ziegler proved that the chambers are in natural bijection with the biconvex sets of the arrangement. Two families of examples of arrangements with a lattice of chambers are simplicial and supersolvable arrangements. For these arrangements, we prove that the chambers correspond to biclosed sets, a weakening of the biconvex property.