Remark on the identities of the grammic monoid with three generators

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.