On maximal order type of the lexicographic product

TL;DR

This note addresses the problem of determining the maximal order type $o(P\cdot Q)$ for the lexicographic product of well partial orders. It corrects a prior claim that $o(P\cdot Q)=o(P)\cdot o(Q)$ in general, presenting Isa Vialard's formula: if $Q$ has $k$ maximal elements and $o(Q)=\delta+m$ with $\delta$ a limit ordinal and $m\ge k$, then $o(P\cdot Q)=o(P)\cdot[\delta+(m-k)]+o(P)\otimes k$, with the special case $k=0$ giving $o(P\cdot Q)=o(P)\cdot o(Q)$. The proof employs cuts of posets, isomorphic copies, and Cantor normal form with Hessenberg arithmetic, proceeding by induction on $o(Q)$. The paper also clarifies the role of maximal elements in shaping the formula and situates the result within broader corrections to the literature, including authorship disclosures related to earlier versions. Overall, the work provides a precise and general method to compute $o(P\cdot Q)$ for wpos, resolving a longstanding ambiguity in this area.

Abstract

In the previously submitted version of this paper, available here for the record, we stated the following : "We give a self-contained proof of Isa Vialard's formula for $o(P\cdot Q)$ where $P$ and $Q$ are wpos. The proof introduces the notion of a cut of partial order, which might be of independent interest." In fact, the argument presented in the paper is wrong and Vialard formula has no known proof. I will try to prove the formula $o(P\cdot Q)=o(P)\cdot o(Q)$ from the [DzSS] paper because I believe that Altman's purported counter-example mentioned in the preprint is incorrect. This statement is written by Mirna Džamonja without consultation with Isa Vialard, who may hold different views. Mirna Džamonja has withdrawn her authorship from the conditionally accepted version of this note (IGPL) on January 20, 2025

Theorems & Definitions (8)

• Claim 2.1
• Claim 2.2
• Theorem 2.3
• Claim 2.5
• Lemma 2.6
• Lemma 2.7
• Claim 2.8
• Claim 3.2