Principal ideals in a plactic monoid always intersect
Daniel Turaev
TL;DR
This paper proves that two principal ideals in a plactic monoid always intersect, establishing that plactic monoids are both left and right reversible for finite rank and the infinite rank. The approach shows that a witness exists for equations of the form $Xu= Xv$ precisely when the contents satisfy $c(u)=c(v)$, and constructs such witnesses using column generators $f_i$ in $\mathcal{C}_n$ and Schensted insertion. It then leverages the Schützenberger involution $\theta$ to relate left and right reversibility in involution monoids, applying this to $P_n$. From the equal-content criterion, the paper deduces that any pair of principal left ideals intersect in $P_n$, which by a standard corollary implies intersection of principal right ideals, hence reversibility; the infinite-rank case $P_\mathbb{N}$ follows by embedding into some $P_n$. The result clarifies the algebraic structure of plactic monoids and has implications for embeddability questions and related combinatorial constructions.
Abstract
This note presents a proof that two principal ideals in a plactic monoid always intersect. Namely, this means that the plactic monoids are both left and right reversible. To the author's knowledge, this result has not yet appeared in the literature studying this monoid. This result holds for both finite rank plactic monoids and the infinite rank plactic monoid.