  We consider the restrictions of Shi arrangements to Weyl cones, their
relations to antichains in the root poset, and their intersection posets. For
any Weyl cone, we provide bijections between regions, flats intersecting the
cone, and antichains of a naturally-defined subposet of the root poset. This
gives a refinement of the parking function numbers via the Poincar\'e
polynomials of the intersection posets of all Weyl cones. Finally, we interpret
these Poincar\'e polynomials as the Hilbert series of two isomorphic graded
rings, one arising from the Varchenko-Gel'fand ring and another, which we call
the order ring since it turns out to be naturally associated to the order
polytope.
