Remark on the identities of the grammic monoid with three generators

TL;DR

This paper investigates the identities of grammic monoids with three generators, showing that despite their quotient relationship with the plactic monoid, they satisfy the same algebraic identities.

Contribution

It proves that grammic and plactic monoids with three generators share the same identities, clarifying their algebraic relationship.

Findings

Grammic monoids with three generators satisfy the same identities as the plactic monoid.

The quotient of the plactic monoid by a specific congruence remains proper.

Despite being a proper quotient, the identities of the two monoids coincide.

Abstract

Grammic monoids have recently been introduced by Christian Choffrut in terms of the action of the free monoid over a fixed ordered alphabet $X$ on the set of rows of Young tableaux filled with elements from $X$ via Schensted's insertion. For $X=\{a,b,c\}$ with $a<b<c$, Choffrut has identified the grammic monoid on $X$ with the quotient of the plactic monoid on $X$ over the congruence generated by the pair $(bacb,cbab)$. Since $cbab\ne bcab$ in the latter monoid, the quotient is proper. We show that, nevertheless, the plactic and the grammic monoids with three generators satisfy the same identities.