2 - Particular classes of ordered sets

Summary

This book contains results such as Dilworth's or Hiraguchi's Theorems (Chapters 4 and 6 respectively) that hold for any ordered set. However, this type of general result is rather rare. The notion of an order, although very restrictive compared to the notion of a relation, actually remains very general, a fact revealed by the huge number of different orders that can be obtained on a set with a small size (more than two million types of order on a set with 10 elements! See Appendix C). Yet in practice, the orders that naturally appear in many contexts most often belong to some particular classes of orders. These classes may be defined in many ways. They are obtained, for instance, by setting the value of a parameter (for instance, orders of dimension 2, studied in Section 6.3), by forbidding the presence of some given configurations (for instance, interval orders mentioned in Example 1.22 and in Section 2.2), by constructing the class by iteration of some given operations on a family of initial orders (for instance, series–parallel orders, defined in Section 2.2). In this chapter, we present some of the most frequent classes of orders, that will be regularly encountered all throughout the book. Although we define them in a unique way here, we will see in the exercises and later in the text that these classes often have several alternative definitions. This explains the fact that they have sometimes appeared independently in various contexts and reinforces their interest.