TR-113: Angle Orders, Regular n-gon Orders and the Crossing Number of a Partial Order

Abstract

A finite poset P(X,<) on a set X={x1, … ,xm} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and
having parallel sides) such that xi<xj if and only if the angular region (regular n-gon) for xi is contained in the
region (regular n-gon) for Xj. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using eq uliateral triangles with the same orientation. This result does not generalize to higher dimensions. We will prove that there is a
partial order of dimension 4 with 14 elements which is not an regular n-gon order regardless of the value of n.
Finally, we prove that partial orders of dimension 3, are regular n-gon orders for n3.