A new approach to the grammic monoid
Marianne Johnson, António Malheiro
TL;DR
The paper provides a new, sequence-based description of the grammic monoid of rank n by linking it to weakly increasing subsequences and a faithful tropical representation into upper triangular tropical matrices. This yields that the grammic monoid and UT_n(T) generate the same semigroup identities, extending previous rank-3 results to all finite ranks and clarifying the infinite-rank behavior. The work also establishes compatibility properties, explores generalized Knuth-like relations and column-based presentations, and poses several foundational open questions about finite presentations and the structure of grammic varieties. Together, these results deepen the connection between grammic monoids, plactic monoids, and tropical algebra, with implications for decidability and identity theory in semigroup varieties.
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.