A new approach to the grammic monoid
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908679423393792 |
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| author | Johnson, Marianne Malheiro, António |
| author_facet | Johnson, Marianne Malheiro, António |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new approach to the grammic monoid Johnson, Marianne Malheiro, António Rings and Algebras 20M07, 20M30, 05E99, 12K10, 16Y60 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. |
| title | A new approach to the grammic monoid |
| topic | Rings and Algebras 20M07, 20M30, 05E99, 12K10, 16Y60 |
| url | https://arxiv.org/abs/2511.22514 |