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Well-Partial Orderings and their Maximal Order Types

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Well-Quasi Orders in Computation, Logic, Language and Reasoning

Part of the book series: Trends in Logic ((TREN,volume 53))

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.

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Notes

  1. 1.

    But see also the note on p. 8.

  2. 2.

    Except for the very rough first attempt in Chap. II, Sect. 4 of [21].

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Acknowledgements

I am grateful to Dick de Jongh for inviting me to Amsterdam in 1973 and introducing me to his work, and to Professor Helmut Pfeiffer for inviting me to Hannover in 1977 and putting to me a problem which led me to the results in this paper; to Professor Gert H.Müller for his active interest in my personal and mathematical welfare since 1974; and, last but by no means least, to my husband and children for their tolerance and support.

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Authors and Affiliations

  1. Hochschule Heilbronn, Max-Planck-Str. 39, D-74081, Heilbronn, Germany

    Diana Schmidt (Emeritus)

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Correspondence to Diana Schmidt .

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Editors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Verona, Verona, Italy

    Peter M. Schuster

  2. Department of Computer Science, Swansea University, Swansea, UK

    Monika Seisenberger

  3. Vakgroep Wiskunde, Ghent University, Ghent, Belgium

    Andreas Weiermann

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Diana Schmidt (2020). Well-Partial Orderings and their Maximal Order Types. In: Schuster, P., Seisenberger, M., Weiermann, A. (eds) Well-Quasi Orders in Computation, Logic, Language and Reasoning. Trends in Logic, vol 53. Springer, Cham. https://doi.org/10.1007/978-3-030-30229-0_13

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