Representing orders on the plane by translating convex figures (Q1114721)

Suppose each figure in a finite collection of disjoint convex plane figures is assigned a single direction of motion not necessarily all the same. We say that the figure B obstructs the figure A if there is a line joining a point of A to a point of B which follows the direction assigned to A. Then a blocking relation is defined such that B blocks A if there is a sequence \(A=A_ 1,...A_ k=B\) with each \(A_{i+1}\) obstructing \(A_ i\). We call the relation m-directional if each of the figures is assigned one of m directions. The following three theorems are proved. 1. There is a one-to-one correspondence between the class of all one- directional blocking relations and the class of all truncated planar lattices. A lattice is planar if its diagram can be drawn without any crossing lines. It is truncated when its greatest and least elements are removed. 2. Every ordered set has a subdivision which can be represented as a two- directional blocking relation. 3. There is an ordered set which has no m-directional representation for any positive integer m.