Free Triangle Orders

We define a new class of ordered sets, called free triangle orders. These are ordered sets represented by a left-to-right ordering on geometric objects contained in a horizontal strip in the plane. The objects are called “free triangles,” and have one vertex on each of the two boundaries of the strip and one vertex in its interior. These ordered sets generalize the classes of trapezoid and triangle orders studied in (Bogart et al., 1998; Ryan, 1998; Ryan, 1999), represented by trapezoids and triangles respectively, contained within a strip in the plane, and are a special case of the orders of tube dimension 2 introduced by (Habib et al., 1992), which are represented by any set of convex bodies contained within a strip in the plane. Our main result is that the class of free triangle orders strictly contains the class of trapezoid orders.