Hyperplane arrangements in preference modeling

A standard representation of a family B of partial orders on a given finite set X is as a set of vertices of a cube. The metric and order structures on B inherited from the cube are often used in applications. In this paper, following Stanley [(1996). Hyperplane arrangements, interval orders, and trees. Proceedings of the National Academy of Sciences of the United States of America, 93, 2620-2625], we represent relations in B by regions and cells of a hyperplane arrangement arising from numerical representations of the partial orders in B. To illustrate this approach, we establish wellgradedness of some families of generalized semiorders. Although the families of linear and weak orders are not well graded, our approach allows the recasting of such concepts as well graded families of sets.