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Introduction to Lattice Theory with Computer Science Applications by Vijay K. Garg

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Chapter 12Interval Orders

12.1 Introduction

This chapter covers weak orders (ranking), semiorders, and interval orders. All of these partial orders are “close” to a total order. The relationship between these orders is as follows. Every weak order is also a semiorder, and every semiorder is also an interval order.

12.2 Weak Order

A weak order, also called ranking, is a slight generalization of a total order.

The term “weak order” is somewhat of a misnomer because a weak order has a lot of order. To allow for easy visualization of such orders, we will use the term “ranking” instead of the more accepted term “weak order.” The set of elements in a ranking which have the same c12-math-0004 value is called a rank. For example, the poset c12-math-0005 shown in Figure 12.1 is a ranking because we can assign , and . Here, and are in the same rank. The difference between a chain and a ranking is that a ...

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