Segment Orders

We study two kinds of segment orders, using definitions first proposed by Farhad Shahrokhi. Although the two kinds of segment orders appear to be quite different, we prove several results suggesting that the are very much the same. For example, we show that the following classes belong to both kinds of segment orders: (1) all posets having dimension at most 3; (2) interval orders; and for n ≥ 3, the standard example Sn of an n-dimensional poset, all 1-element and (n− 1)-element subsets of {1,2, . . . , n}, partially ordered by inclusion. Moreover, we also show that, for each d ≥ 4, almost all posets having dimension d belong to neither kind of segment orders. Motivated by these observations, it is natural to ask whether the two kinds of segment orders are distinct. This problem is apparently very difficult, and we have not been able to resolve it completely. The principal thrust of this paper is the development of techniques and results concerning the properties that must hold, should the two kinds of segment orders prove to be the same. We also derive equivalent statements, one version of which is a stretchability question involving certain sets of pseudoline arrangements. We conclude by proving several facts about continuous universal functions that would transfer segment orders of the first kind into segments orders of the second kind.