Orders and Abstract Reduction Relations

Abstract

The theory of Gröbner bases deals with ideals in polynomial rings and is thus part of commutative algebra. However, the concept of binary relations, and in particular, of orders, is instrumental in making Gröbner basis theory work. This chapter provides the necessary results by discussing binary relations on an abstract set M. Our treatment centers around the study of various kinds of finiteness properties such as well-foundedness. These properties will later be used in a number of ways; eventually, however, their relevance lies in the fact that they provide termination proofs for certain algorithms. We will frequently encounter the axiom of choice which we discuss in an introductory section to this chapter.