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Towards Better: A motivated introduction to better-quasi-orders

Yann Pequignot

Abstract

The well-quasi-orders (WQO) play an important role in various fields such as Computer Science, Logic or Graph Theory. Since the class of WQOs lacks closure under some important operations, the proof that a certain quasi-order is WQO consists often of proving it enjoys a stronger and more complicated property, namely that of being a better-quasi-order (BQO). Several articles contains valuable introductory material to the theory of BQOs. However, a textbook entitled "Introduction to better-quasi-order theory" is yet to be written. Here is an attempt to give a motivated and self-contained introduction to the deep concept defined by Nash-Williams that we would expect to find in such a textbook.

Towards Better: A motivated introduction to better-quasi-orders

Abstract

The well-quasi-orders (WQO) play an important role in various fields such as Computer Science, Logic or Graph Theory. Since the class of WQOs lacks closure under some important operations, the proof that a certain quasi-order is WQO consists often of proving it enjoys a stronger and more complicated property, namely that of being a better-quasi-order (BQO). Several articles contains valuable introductory material to the theory of BQOs. However, a textbook entitled "Introduction to better-quasi-order theory" is yet to be written. Here is an attempt to give a motivated and self-contained introduction to the deep concept defined by Nash-Williams that we would expect to find in such a textbook.

Paper Structure

This paper contains 18 sections, 33 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Well-quasi-orders
  4. Good versus bad sequences
  5. Subsets and downsets
  6. Regular sequences
  7. Sequences of subsets
  8. Better-quasi-orders
  9. Towards better
  10. Super-sequences
  11. Index sets for super-sequences
  12. Sub-front and sub-super-sequences
  13. Multi-sequences
  14. Iterated powerset, determinacy of finite games
  15. Equivalence
  16. ...and 3 more sections

Key Result

Proposition 2.2

For a quasi-order $Q$, the following conditions are equivalent.

Figures (5)

  • Figure 1: Rado's poset $\mathfrak{R}$.
  • Figure 2: Pictures of fronts
  • Figure 3: Constructing a multi-sequence by stringing strategies together.
  • Figure 4: Stringing strategies together.
  • Figure 5: Copying and shift.

Theorems & Definitions (87)

  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.6
  • proof
  • Definition 2.8
  • Proposition 2.9
  • proof
  • proof
  • ...and 77 more