Remark on the identities of the grammic monoid with three generators

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 $cbab\mathrm{\ne }bcab$ 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.