A new approach to the grammic monoid

TL;DR

This paper introduces a novel description of the grammic monoid using weakly increasing subsequences, establishes its isomorphism with a tropical matrix representation, and explores its algebraic identities across different ranks.

Contribution

It provides a new characterization of the grammic monoid, links it to tropical matrix representations, and extends known identity results to infinite rank cases.

Findings

The grammic monoid is isomorphic to the image of a tropical representation.

Rank 3 grammic monoid satisfies the same identities as 3x3 upper triangular tropical matrices.

Infinite rank grammic monoid satisfies no non-trivial semigroup identities.

Abstract

We give an alternative description of the grammic monoid in terms of weakly increasing subsequences. Specifically, we show that words $u,v$ in the generators $\{1,\ldots, n\}$ determine the same element of the grammic monoid of rank $n$ if and only if for all $1 \leq p \leq q$, the maximum length of a weakly increasing subsequence on alphabet $\{p,\ldots, q\}$ is the same in $u$ and $v$. Our proof makes use of a particular tropical representation of the plactic monoid determined by such sequences: we demonstrate that the grammic monoid is isomorphic to the image of this representation, and (by applying a result of the first author and Kambites) immediately deduce that the grammic monoid of rank $n$ satisfies exactly the same semigroup identities as the monoid of $n \times n$ upper triangular tropical matrices. This gives a partial generalisation of a result of Volkov, who has shown that the grammic monoid of rank $3$ satisfies exactly the same semigroup identities as the plactic monoid of rank $3$ which in turn is known (by applying a result of the first author and Kambites) to satisfy the exactly the same semigroup identities as the monoid of $3 \times 3$ upper triangular tropical matrices. Furthermore, we find that the grammic monoid of infinite rank does not satisfy any non-trivial semigroup identity, and demonstrate that the grammic congruence satisfies some useful compatibility properties.