On maximal order type of the lexicographic product

TL;DR

This paper discusses the maximal order type of the lexicographic product of well partial orders, clarifying the status of Vialard's formula and presenting a corrected approach to its proof.

Contribution

The paper critically examines Vialard's formula for the order type of lexicographic products and attempts to establish a correct proof based on existing literature.

Findings

Vialard's formula for $o(P\cdot Q)$ is unproven and disputed.

The paper offers a new approach to prove $o(P\cdot Q)=o(P)\cdot o(Q)$.

Authors withdraw authorship due to uncertainties in the proof.

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 D\v{z}amonja without consultation with Isa Vialard, who may hold different views. Mirna D\v{z}amonja has withdrawn her authorship from the conditionally accepted version of this note (IGPL) on January 20, 2025