On Numerical Semigroups with Fixed Quotient
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913129186721792 |
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| author | Ojeda, Ignacio Rosales, José Carlos |
| author_facet | Ojeda, Ignacio Rosales, José Carlos |
| contents | Let $Δ$ be a numerical semigroup and let $d\ge 2$ be an integer. We study the fiber of the quotient map \(S\mapsto S/d\) over $Δ$. We describe its elements as semigroups of the form $\langle X\rangle+dΔ$, for suitable finite sets $X\subseteqΔ$, and then analyze explicit and computable regions of this fiber. In particular, we introduce a family $Δ_d(a)$ of multiples with prescribed quotient and compute its generators, classical invariants, Apéry sets, and presentations. We also show that this construction preserves Wilf's inequality and controls the depth. Finally, we introduce the $\mathcal{M}_d(Δ)$-rank, determine its maximal value in the fiber, relate it to the ordinary embedding dimension, characterize the rank-one elements, and give closed formulas for their Frobenius-type invariants and pseudo-Frobenius numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14775 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Numerical Semigroups with Fixed Quotient Ojeda, Ignacio Rosales, José Carlos Commutative Algebra Primary 20M14, Secondary 11D07, 20M25 Let $Δ$ be a numerical semigroup and let $d\ge 2$ be an integer. We study the fiber of the quotient map \(S\mapsto S/d\) over $Δ$. We describe its elements as semigroups of the form $\langle X\rangle+dΔ$, for suitable finite sets $X\subseteqΔ$, and then analyze explicit and computable regions of this fiber. In particular, we introduce a family $Δ_d(a)$ of multiples with prescribed quotient and compute its generators, classical invariants, Apéry sets, and presentations. We also show that this construction preserves Wilf's inequality and controls the depth. Finally, we introduce the $\mathcal{M}_d(Δ)$-rank, determine its maximal value in the fiber, relate it to the ordinary embedding dimension, characterize the rank-one elements, and give closed formulas for their Frobenius-type invariants and pseudo-Frobenius numbers. |
| title | On Numerical Semigroups with Fixed Quotient |
| topic | Commutative Algebra Primary 20M14, Secondary 11D07, 20M25 |
| url | https://arxiv.org/abs/2605.14775 |