On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

Mirna Dzamonja, Sylvain Schmitz, Philippe Schnoebelen

Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)

Abstract

We investigate the ordinal invariants height, length, and width of well quasi orders (WQO), with particular emphasis on width, an invariant of interest for the larger class of orders with finite antichain condition (FAC). We show that the width in the class of FAC orders is completely determined by the width in the class of WQOs, in the sense that if we know how to calculate the width of any WQO then we have a procedure to calculate the width of any given FAC order. We show how the width of WQO orders obtained via some classical constructions can sometimes be computed in a compositional way. In particular this allows proving that every ordinal can be obtained as the width of some WQO poset. One of the difficult questions is to give a complete formula for the width of Cartesian products of WQOs. Even the width of the product of two ordinals is only known through a complex recursive formula. Although we have not given a complete answer to this question we have advanced the state of knowledge by considering some more complex special cases and in particular by calculating the width of certain products containing three factors. In the course of writing the paper we have discovered that some of the relevant literature was written on cross-purposes and some of the notions re-discovered several times. Therefore we also use the occasion to give a unified presentation of the known results.

Original languageEnglish
Title of host publicationWell quasi-orders in computation, logic, language and reasoning
Subtitle of host publicationA unifying concept of proof theory, automata theory, formal languages and descriptive set theory
EditorsPeter Schuster, Monika Seisenberger, Andreas Weiermann
PublisherSpringer
Chapter2
Pages29-54
Number of pages26
ISBN (Print)978-3-030-30229-0
DOIs
Publication statusPublished - 2 Jan 2020

Publication series

NameTrends in Logic
Volume53
ISSN (Print)1572-6126
ISSN (Electronic)2212-7313

Keywords

  • Ordinal invariants
  • WQO
  • Width of WQO

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