On maximal order type of the lexicographic product

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 Mirna Džamonja has withdrawn her authorship from the conditionally accepted version of this note (IGPL) on January 20, 2025

An incorrect attribution to Abraham and Bonnet of the formula $o(P\cdot Q)=o(P)\cdot o(Q)$ was given in arxiv.org/abs/1711.00428 in MATH/LO