On quotients of numerical semigroups for almost arithmetic progressions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911578863960064 |
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| author | Liu, Feihu |
| author_facet | Liu, Feihu |
| contents | Let $\langle A\rangle$ be the numerical semigroup generated by relatively prime positive integers $\{a_1,a_2,...,a_n\}$. The quotient of $\langle A\rangle$ with respect to a positive integer $p$ is defined by $\frac{\langle A\rangle}{p}=\{x\in \mathbb{N} \mid px\in \langle A\rangle\}$. The quotient $\frac{\langle A\rangle}{p}$ is known to be a semigroup but is hard to study. When $p$ is a positive divisor of $a_1$, we reduce the computation of the Apéry set of $\frac{a_1}{p}$ in $\frac{\langle A\rangle}{p}$ to a simple minimization problem. This allow us to obtain closed formulas of the Frobenius number of the quotient for some special numerical semigroups. These includes the cases when $\langle A\rangle$ is the almost arithmetic progressions, the almost arithmetic progressions with initial gaps, etc. In particular, we partially solve an open problem proposed by A. Adeniran et al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06096 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On quotients of numerical semigroups for almost arithmetic progressions Liu, Feihu Number Theory Combinatorics Let $\langle A\rangle$ be the numerical semigroup generated by relatively prime positive integers $\{a_1,a_2,...,a_n\}$. The quotient of $\langle A\rangle$ with respect to a positive integer $p$ is defined by $\frac{\langle A\rangle}{p}=\{x\in \mathbb{N} \mid px\in \langle A\rangle\}$. The quotient $\frac{\langle A\rangle}{p}$ is known to be a semigroup but is hard to study. When $p$ is a positive divisor of $a_1$, we reduce the computation of the Apéry set of $\frac{a_1}{p}$ in $\frac{\langle A\rangle}{p}$ to a simple minimization problem. This allow us to obtain closed formulas of the Frobenius number of the quotient for some special numerical semigroups. These includes the cases when $\langle A\rangle$ is the almost arithmetic progressions, the almost arithmetic progressions with initial gaps, etc. In particular, we partially solve an open problem proposed by A. Adeniran et al. |
| title | On quotients of numerical semigroups for almost arithmetic progressions |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2312.06096 |