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On the Width of the Cartesian Product of Ordinals

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Abstract

The ordinal invariants, i.e., maximal order type, height, and width, are measures of a well quasi-ordering (wqo) based on the ordinal rank of the trees of its bad sequences, strictly decreasing sequences, and antichain sequences, respectively. Complex wqos are often built from simpler wqos through basic constructions such as disjoint sum, direct sum, cartesian product, and higher-order constructions like powerset or sequences. One main challenge is to compute the ordinal invariants of such wqos compositionally. This article focuses on the width of the cartesian product of wqos, for which no general formula is known. The particular case of the cartesian product of two ordinals has already been solved by Abraham in 1987, using the methods of residuals. We introduce a new method to get lower bounds on width, and apply it to the width of the cartesian product of finitely many ordinals, thus generalizing Abraham’s result. Finally, we leverage this result to compute the width of a generic family of elementary wqos that is closed under cartesian product.

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Acknowledgements

The research described in this article started with a master internship supervized by Ph. Schnoebelen. It has also benefited from numerous discussions and suggestions from M. Džamonja, S. Halfon, and S. Schmitz. We also thank the reviewers for their thorough feedback.

Funding

This work was supported by ANR project BRAVAS (grant ANR-17-CE40-0028) and the École Polytechnique AMX program.

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  1. LMF, Université Paris-Saclay, ENS Paris-Saclay, CNRS, Gif-sur-Yvette, France

    Isa Vialard

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Vialard, I. On the Width of the Cartesian Product of Ordinals. Order (2024). https://doi.org/10.1007/s11083-024-09668-8

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