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.
Mirna Džamonja thanks the Leverhulme Trust for a Research Fellowship for the academic year 2014/2015, the Simons Foundation for a Visiting Fellowship in the Autumn of 2015 and the Isaac Newton Institute in Cambridge for their support during the HIF programme in the Autumn of 2015, supported by EPSRC Grant Number EP/K032208/1. She most gratefully acknowledges the support of the Institut d’histoire et de philosophie des sciences et des techniques (IHPST) at University Paris 1, where she is as an Associate Member. The three authors thank the London Mathematical Society for their support through a Scheme 3 Grant.
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Notes
- 1.
We adopted the definition from Laver [12].
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Džamonja, M., Schmitz, S., Schnoebelen, P. (2020). On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders. 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_2
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