The Schützenberger groups and maximal subgroups of tropical matrices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909457885167616 |
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| author | Aird, Thomas |
| author_facet | Aird, Thomas |
| contents | We classify the Schützenberger groups of the category of matrices over the tropical semiring, $M(\mathbb{T})$, in doing so, we obtain a classification for the Schützenberger groups of the semigroupoid of matrices over the finitary tropical semiring, $M(\mathbb{F}\mathbb{T})$. We then classify the maximal subgroups of the monoid of $n \times n$ matrices over the tropical semiring, $M_n(\mathbb{T})$, for all $n \in \mathbb{N}$; generalising a result in the literature and correcting an erroneous proof. We proceed to show that for some $n \in \mathbb{N}$ there exists a group which appears as a Schützenberger group of $M_n(\mathbb{T})$ but does not appear as a maximal subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Schützenberger groups and maximal subgroups of tropical matrices Aird, Thomas Group Theory Rings and Algebras 20H99 (Primary) 16Y60, 12K10, 20M10 We classify the Schützenberger groups of the category of matrices over the tropical semiring, $M(\mathbb{T})$, in doing so, we obtain a classification for the Schützenberger groups of the semigroupoid of matrices over the finitary tropical semiring, $M(\mathbb{F}\mathbb{T})$. We then classify the maximal subgroups of the monoid of $n \times n$ matrices over the tropical semiring, $M_n(\mathbb{T})$, for all $n \in \mathbb{N}$; generalising a result in the literature and correcting an erroneous proof. We proceed to show that for some $n \in \mathbb{N}$ there exists a group which appears as a Schützenberger group of $M_n(\mathbb{T})$ but does not appear as a maximal subgroup. |
| title | The Schützenberger groups and maximal subgroups of tropical matrices |
| topic | Group Theory Rings and Algebras 20H99 (Primary) 16Y60, 12K10, 20M10 |
| url | https://arxiv.org/abs/2501.09501 |