Lengths of Bounded-Rank Linear-Order WQOs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A countable linear order is scattered if it contains no copy of $\mathbb Q$. Use the Hausdorff hierarchy $\mathcal H_0$ of finite orders; for $\beta>0$, $\mathcal H_\beta$ consists of finite sums of orders $\sum_{n\in\mathbb Z}L_n$ with each $L_n\in\bigcup_{\gamma<\beta}\mathcal H_\gamma$. The Hausdorff rank is the least $\beta$ with $L\in\mathcal H_\beta$. Identify orders $L,K$ when each order-embeds into the other (equimorphism). For a countable ordinal $\alpha$, let $L_\alpha$ be the resulting partial order of scattered orders of rank less than $\alpha$, ordered by embeddability. A quasi-order is wqo if each infinite sequence has increasing indices $i<j$ with $x_i\le x_j$. For a wqo partial order $W$, define $o(W)$ as the supremum of the ordinal order types of its linear extensions.

Statement recorded in the supplied source.

Determine the maximal order type $o(L_\alpha)$ for every countable ordinal $\alpha$. [S2]

Short English statement

Compute the length of the embeddability ordering of countable linear orders below each Hausdorff-rank bound, after identifying mutually embeddable orders.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7500122 (AMR-074-0122). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Antonio Montalbán, Open Questions in Reverse Mathematics, author preprint (17 August 2010), Question 22, Section 4.2, p. 9 and Definition 4.2, p. 8. https://math.berkeley.edu/~antonio/papers/questionsRM.pdf.