On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

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.

On the cartesian product of well-orderings

“…These formulas can be seen as a reformulation of tree rank computation (see Section 2.3. of [DSS20] and the references therein). We can use them to recursively compute the invariants of A: this is called the method of residuals.…”

“…Section 2 introduces definitions, notations and recalls known results, mostly following [DSS20]. Section 3 proves intermediary results on lower bounds on the width that lay the ground for future proofs.…”

“…Since T can be infinitely branching, its rank is a possibly infinite ordinal. For a more general definition of the rank of any well-founded partial-order set, see Section 2.2 of [DSS20].…”

“…We define the disjoint sum , lexicographic sum +, cartesian product × and direct product • as in [DSS20].…”

“…A recent article by Dzamonja et al ( [DSS20]) shows that we do not always know how to compute the width of wqos, even in the apparently simple case of a cartesian product. This problem is unfortunate since the cartesian product is the most common and basic data structure in mathematics and computer science.…”