Well-Partial Orderings and their Maximal Order Types

Abstract

Combinatorial theorists have for some time been showing that certain partial orderings are well-partial-orderings (w.p.o.’s). De Jongh and Parikh showed that w.p.o.’s are just those well-founded partial orderings which can be extended to a well-ordering of maximal order type; we call the ordinal thus obtained the maximal order type of the w.p.o. In this paper we calculate, in terms of a system of notations due to Schütte [24], the maximal order types of the w.p.o.’s investigated in Higman [11], and give upper bounds for the maximal order types of the w.p.o.’s investigated in Kruskal [13] and Nash-Williams [16]. As a by-product and an application of de Jongh and Parikh’s work, we give new and easier proofs of Higman’s, Kruskal’s and Nash–Williams’ theorems that the partial orderings considered are indeed w.p.o.’s. We also apply our results to the theory of ordinal notations.