On the ABK Conjecture, alpha-well Quasi Orders and Dress-Schiffels product
Uri Abraham, Robert Bonnet, Mirna Džamonja, Maurice Pouzet
TL;DR
The paper tackles the Abraham–Bonnet–Kubi\'s (ABK) conjecture that every $wqo$ is a countable union of $bqo$ by developing the theory of $\,\sigma$-bqo and a spectrum of $\alpha$-wqo notions. It proves strong closure properties for $\sigma$-bqo under a range of operations, notably including the Dress-Schiffels product, and establishes transfer principles between $P^\alpha$, $P^{<\alpha}$, and $I(P^{<\alpha})$, alongside a Hausdorff-style classification of $\alpha$-FAC orders. The results provide partial progress toward ABK by showing that potential counterexamples must arise from new ideas beyond familiar operations on $wqo$ posets and by clarifying the structure of $\alpha$-wqo/spines and their built-up closures. Overall, the work advances understanding of how $wqo$ can be decomposed into $bqo$-like components and outlines a framework for further exploration of the ABK conjecture’s boundaries.
Abstract
The following is a 2008 conjecture of Abraham, Bonnet and Kubiś: [ABK Conjecture] Every well quasi order (wqo) is a countable union of better quasi orders (bqo). We obtain a partial progress on the conjecture, by showing that the class of orders that are a countable union of better quasi orders (sigma-bqo) is closed under various operations. These include diverse products, such as the Dress-Shieffels product. We develop various properties of the latter product. In relation with the main question, we explore the class of alpha-wqo for countable ordinals alpha and obtain several closure properties and a Hausdorff-style classification theorem. Our main contribution is the discovery of various properties of sigma-bqos and ruling out potential counterexamples to the ABK Conjecture.