Non-Unit Free Triangle Orders

Abstract

A Free Triangle order is a partially ordered set in which every element can be represented by a triangle. All triangles lie between two parallel baselines, with each triangle intersecting each baseline in exactly one point. Two elements in the partially ordered set are incomparable if and only if their corresponding triangles intersect. A unit free triangle order is one with such a representation in which all triangles have the same area. In this paper, we present an example of a non-unit free triangle order.Key words geometric representation · free triangle order · unit vs. proper.The notion of a free triangle representation of a partially ordered set was first introduced by Laison [3] as a generalization of the ideas of interval and trapezoid representations. A free triangle representation assigns a triangle to each element of a partially ordered set, with all triangles having one vertex on each of two parallel baselines and a third 'free' vertex between the two baselines. In this paper, we examine and separate the classes of unit free triangle orders (those posets having a representation in which all triangles have the same area) and proper free triangle orders (those posets having a representation in which no free triangle contains another).

A Finite Non-Unit Free Triangle Order

“…As in [1], we will make heavy use of the following observation about unit free triangle orders. THEOREM 1.…”

“…THEOREM 1. (Balof and Bogart [1]) The horizontal distance is the same for all triangles in a unit triangle representation.…”

“…THEOREM 2. (Balof and Bogart [1]) Let (X, P) be a free triangle order, and let s ∈ X. Then (X, P) has a free triangle representation in which the hypotenuse for s is perpendicular to the baselines.…”

“…As shown in [1], we may drop the element x 12 or x 42 from S ++ 3 and achieve an order with many of the same necessary 'forcing' properties in free triangle representation (we chose not to do that here for ease of notation and exposition). It is unknown whether there is a fundamentally smaller non-unit free triangle order (i.e., is there a less cumbersome way to do all of this forcing).…”

“…For trapezoid orders, unit = proper, that is, there are orders that are proper but have no unit representation (see [3] for an example). In [1], we showed that unit =proper for free triangle orders by finding an example of an infinite free triangle order with a proper representation, but no unit representation. In this paper we will construct a more complicated but finite non-unit free triangle order.…”