On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

TL;DR

The paper investigates the ordinal invariants height $\bm h$, width $\bm w$, and maximal order type $\bm o$ for well quasi-orders (WQO), with a focus on width for FAC posets. It establishes that the FAC width is determined by the WQO width, and develops residual and game-based methods to compute these invariants, yielding compositional formulas for lexicographic sums, disjoint sums, and many finite constructions, while Cartesian products remain largely open. The work also unifies disparate results across heights, linearisations, and downward-closed sets, and provides complete descriptions for several common WQOs and their finite-label extensions, including multisets, sequences, and trees. It further discusses limits via Rado’s structure to illustrate the boundaries of invariant-based characterisations. Overall, the paper advances the understanding of width computations in WQOs and FAC posets, clarifying when and how the three invariants interact and highlighting key open questions in Cartesian products.

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. ERRATUM:We incorrectly claimed in Lemma 4.4(1) the formula $o(P\cdot Q)=o(P)\cdot o(Q)$ for wpos $P$ and $Q$ and incorrectly attributed it to Abraham and Bonnet. We incorrectly claimed in 4.4(2) that the formula for $h(P\cdot Q)$ was due to Abraham and Bonnet.

Theorems & Definitions (64)

• Lemma 2.1
• proof : Proof of 1
• proof : Proof of 2
• Lemma 2.2
• proof
• Theorem 2.3: Dushnik-Miller
• proof
• Theorem 2.4: abcdzt
• Lemma 2.5
• proof