Linear-Interval Dimension and PI Orders

A PI graph G is the intersection graph of a family of triangles ABC between two distinct parallel lines L1 and L2, such that A is on L1 and is on L2. We study the orders defined by transitive orientations of the complement of G, the PI orders. We describe a characterization for such orders in terms of a special order dimension called linear-interval dimension. We show that the linear-interval dimension of an order is a comparability invariant, which generalizes the well-known result that the interval dimension is a comparability invariant.