Generating sets, presentations, and growth of tropical matrix monoids
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909605068537856 |
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| author | Aird, Thomas |
| author_facet | Aird, Thomas |
| contents | We construct minimal and irredundant generating sets for a family of submonoids of the monoid of $n \times n$ upper triangular matrices over a commutative semiring. We show that the monoid of $n \times n$ matrices over the tropical integers, $M_n(\mathbb{Z}_\mathrm{max})$, is finitely generated if and only if $n \leq 2$, and finitely presented if and only if $n = 1$. Minimal and irredundant generating sets are explicitly constructed when $n \leq 3$. We then construct a presentation for the monoid of $n \times n$ upper triangular matrices over the tropical integers, $UT_n(\mathbb{Z}_\mathrm{max})$, demonstrating that it is finitely presented for all $n \in \mathbb{N}$. Finally, we establish upper bounds on the polynomial degree of the growth function of finitely generated subsemigroups of the monoid of $n \times n$ matrices over a bipotent semiring and show that these bounds are sharp for the tropical semiring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_12166 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generating sets, presentations, and growth of tropical matrix monoids Aird, Thomas Rings and Algebras 20M10 (Primary) 16Y60, 15A80 (Secondary) We construct minimal and irredundant generating sets for a family of submonoids of the monoid of $n \times n$ upper triangular matrices over a commutative semiring. We show that the monoid of $n \times n$ matrices over the tropical integers, $M_n(\mathbb{Z}_\mathrm{max})$, is finitely generated if and only if $n \leq 2$, and finitely presented if and only if $n = 1$. Minimal and irredundant generating sets are explicitly constructed when $n \leq 3$. We then construct a presentation for the monoid of $n \times n$ upper triangular matrices over the tropical integers, $UT_n(\mathbb{Z}_\mathrm{max})$, demonstrating that it is finitely presented for all $n \in \mathbb{N}$. Finally, we establish upper bounds on the polynomial degree of the growth function of finitely generated subsemigroups of the monoid of $n \times n$ matrices over a bipotent semiring and show that these bounds are sharp for the tropical semiring. |
| title | Generating sets, presentations, and growth of tropical matrix monoids |
| topic | Rings and Algebras 20M10 (Primary) 16Y60, 15A80 (Secondary) |
| url | https://arxiv.org/abs/2201.12166 |