We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order.

我们将面部弱阶从有限 Coxeter 群扩展到中心超平面排列。面部弱阶将超平面排列的区域的偏序扩展到其所有面部。我们提供了中央排列的面部弱顺序的四个非平凡等价的定义：（1）利用面部是区域偏序中的间隔这一事实，（2）通过描述其覆盖关系，（3）使用协向量相应的定向拟阵，以及 (4) 使用与相应带位的几何形状密切相关的某些法向量集。使用这些等效描述，我们表明当区域的偏集是格子时，面部弱阶是格子。在单纯排列的情况下，我们进一步证明这个格是半分布的，并描述了它的连接不可约元素。