A note on Garside monoids and Braces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866912124511453184 |
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| author | Chouraqui, Fabienne |
| author_facet | Chouraqui, Fabienne |
| contents | A left brace is a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is an abelian group, $(\mathcal{B},\cdot)$ is a group, and there is a left-distributivity-like axiom that relates between the two operations in $\mathcal{B}$. In analogy with a left brace, we define a left $\mathscr{M}$-brace to be a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is a commutative monoid, $(\mathcal{B},\cdot)$ is a monoid, and the axiom of left distributivity holds. A lcm-monoid $M$ is a left-cancellative monoid such that $1$ is the unique invertible element in $M$, and every pair of elements in $M$ admit a lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids. We show that every lcm-monoid induces a left $\mathscr{M}$-brace. Furthermore, we show that every Gaussian group induces a partial left brace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_11674 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A note on Garside monoids and Braces Chouraqui, Fabienne Group Theory Rings and Algebras A left brace is a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is an abelian group, $(\mathcal{B},\cdot)$ is a group, and there is a left-distributivity-like axiom that relates between the two operations in $\mathcal{B}$. In analogy with a left brace, we define a left $\mathscr{M}$-brace to be a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is a commutative monoid, $(\mathcal{B},\cdot)$ is a monoid, and the axiom of left distributivity holds. A lcm-monoid $M$ is a left-cancellative monoid such that $1$ is the unique invertible element in $M$, and every pair of elements in $M$ admit a lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids. We show that every lcm-monoid induces a left $\mathscr{M}$-brace. Furthermore, we show that every Gaussian group induces a partial left brace. |
| title | A note on Garside monoids and Braces |
| topic | Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2106.11674 |