Principal ideals in a plactic monoid always intersect

TL;DR

This paper proves that any two principal ideals in a plactic monoid always intersect, establishing the monoid's left and right reversibility, a result not previously documented in the literature.

Contribution

It introduces the first known proof that all principal ideals in a plactic monoid intersect, demonstrating the monoid's reversibility property.

Findings

Principal ideals in a plactic monoid always intersect

Plactic monoids are both left and right reversible

Result applies to finite and infinite rank monoids

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.