On proportionally modular numerical semigroups that are generated by arithmetic progressions
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| Format: | Preprint |
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2020
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| _version_ | 1866913573799723008 |
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| author | Elizeche, Edgar Federico Tripathi, Amitabha |
| author_facet | Elizeche, Edgar Federico Tripathi, Amitabha |
| contents | A numerical semigroup is a submonoid of ${\mathbb Z}_{\ge 0}$ whose complement in ${\mathbb Z}_{\ge 0}$ is finite. For any set of positive integers $a,b,c$, the numerical semigroup $S(a,b,c)$ formed by the set of solutions of the inequality $ax \bmod{b} \le cx$ is said to be proportionally modular. For any interval $[α,β]$, $S\big([α,β]\big)$ is the submonoid of ${\mathbb Z}_{\ge 0}$ obtained by intersecting the submonoid of ${\mathbb Q}_{\ge 0}$ generated by $[α,β]$ with ${\mathbb Z}_{\ge 0}$.
For the numerical semigroup $S$ generated by a given arithmetic progression, we characterize $a,b,c$ and $α,β$ such that both $S(a,b,c)$ and $S\big([α,β]\big)$ equal $S$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_01527 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On proportionally modular numerical semigroups that are generated by arithmetic progressions Elizeche, Edgar Federico Tripathi, Amitabha Number Theory 20M14, 20M30 A numerical semigroup is a submonoid of ${\mathbb Z}_{\ge 0}$ whose complement in ${\mathbb Z}_{\ge 0}$ is finite. For any set of positive integers $a,b,c$, the numerical semigroup $S(a,b,c)$ formed by the set of solutions of the inequality $ax \bmod{b} \le cx$ is said to be proportionally modular. For any interval $[α,β]$, $S\big([α,β]\big)$ is the submonoid of ${\mathbb Z}_{\ge 0}$ obtained by intersecting the submonoid of ${\mathbb Q}_{\ge 0}$ generated by $[α,β]$ with ${\mathbb Z}_{\ge 0}$. For the numerical semigroup $S$ generated by a given arithmetic progression, we characterize $a,b,c$ and $α,β$ such that both $S(a,b,c)$ and $S\big([α,β]\big)$ equal $S$. |
| title | On proportionally modular numerical semigroups that are generated by arithmetic progressions |
| topic | Number Theory 20M14, 20M30 |
| url | https://arxiv.org/abs/2011.01527 |