Abstract
Starting from well-quasi-orders (wqos), we motivate step by step the introduction of the complicated notion of better-quasi-order (bqo). We then discuss the equivalence between the two main approaches to defining bqo and state several essential results of bqo theory. After recalling the rôle played by the ideals of a wqo in its bqoness, we give a new presentation of known examples of wqos which fail to be bqo. We also provide new forbidden pattern conditions ensuring that a quasi-order is a better quasi-order.
Raphaël Carroy was supported by FWF Grants P28153 and P29999. Yann Pequignot gratefully acknowledges the support of the Swiss National Science Foundation (SNF) through grant \(P2LAP2\_164904\).
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Notes
- 1.
The minor relations on finite graphs, proved to be wqo by Robertson and Seymour [32], is to our knowledge the only naturally occurring wqo which is not yet known to be bqo.
- 2.
where \(\omega /m\) denotes the set \(\{n\in \omega \mid m<n\}\).
- 3.
The reader who remains unconvinced can try to prove that the partial order \((3,=)\) satisfies this property.
- 4.
See the introduction of [21].
- 5.
As treated for instance in any introduction to set theory.
- 6.
The only notable exception is \([\omega ]^1\) where both \(\mathrel {\lhd }\) and its complement are actually transitive. If F is a front on \(X=\{x_0,x_1,x_2,\ldots \}\), \(Y=\{x_3,x_4,x_5,\ldots \}\) and if \(s,t,u\in F\) are such that \(s\sqsubset \{x_0\} \cup Y\), \(|s|\ge 2\), \(t\sqsubset \{x_1,x_2\} \cup Y\), \(|t|\ge 2\) and \(u\sqsubset Y\). Then \(s\mathrel {\lhd }u\), while neither \(s\mathrel {\lhd }t\) nor \(t\mathrel {\lhd }u\).
- 7.
Suppose that \(\mathcal {B}\) is a basis for the set of bqos on \(\omega \), i.e. \(B\subseteq 2^{\omega \times \omega }\) and for every qo Q on \(\omega \) we have Q is bqo iff for no \(B\in \mathcal {B}\) there exists an embedding from B to Q. Then \(\mathcal {B}\) is not analytic. Otherwise
$$ Q\text { is } \textsc {bqo}{} \quad \longleftrightarrow \quad \text {there exists no }B\in \mathcal {B} \text { such that }B\text { embeds in }Q $$is a co-analytic definition of the set of bqos on \(\omega \), a contradiction with Marcone’s Theorem.
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Carroy, R., Pequignot, Y. (2020). Well, Better and In-Between. 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_1
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