The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept

Results from the rich and well-developed theory of well-quasi-ordering have often been rediscovered and republished. The purpose of this paper is to describe this intriguing subject. To illustrate the theory, here are two definitions and an elementary result. A partially ordered set is called well-partially-ordered if every subset has at least one, but only a finite number, of minimal elements. For sequences s and t, we define s < t if some subsequence of t majorizes s term by term. Then the space of all finite sequences over a well-partially-ordered set is itself well-partially-ordered.