Well-Graded Families of Relations

Abstract

We have seen in Chapter 3 that wg-families of sets offer a systematic way of representing oriented media in which the sets of the family stand for the states of the medium, and the addition or removal of a given point from any set stand for a token or its reverse. The representation results are formulated by Theorems 3.3.3 and 3.3.4. The purpose of this chapter is to establish that several well-known families of relations, such as the partial orders, the biorders, the interval orders and the semiorders are well-graded, and can thus be represented by media. We also discuss another family, the almost connected orders or ac-orders, which are well-graded if and only if the size of their ground set does not exceed four, offering thus a revealing counterexample.