Summary

Starting with the general abstract definition of a relation, the various sorts of order relations are described and defined, illustrated by many examples. The notion of order isomorphism is introduced. Lattices and Boolean algebras are defined. Examples of these are given and some simple properties derived. Section 3.2 is not a prerequisite for later chapters, although there is reference in Chapter 5 to some of its results.

The reader is presumed to have some experience with abstract algebraic ideas. There is no dependence on the results in Chapters 1 and 2, but in some examples ideas from these chapters are used.

Order relations and ordered sets

The notion of a relation is fundamental in mathematics. Like the notion of a set, it is extremely general and consequently it crops up everywhere. We shall start from the beginning, with the broadest definition, but before we do that, let us observe that there are three kinds of relation which are particularly important, namely, functions, equivalence relations and order relations. Every student of mathematics should know what a function is and how central is the role played by functions in all branches of mathematics. Also, equivalence relations should be familiar although they are perhaps less pervasive. The idea of an order relation, as a general notion, is perhaps less well known, since much of mathematics needs to refer only to particular order relations and does not need to use the general notion or its properties.