{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-complexity-of-a-numerical-semigroup","title":"The complexity of a numerical semigroup","arxiv_id":"2202.00920","date":"2022-02-02","proceeding":null,"authors":["J. I. García-García","M. A. Moreno-Frías","J. C. Rosales","A. Vigneron-Tenorio"],"abstract":"Let $S$ and $\\Delta$ be numerical semigroups. A numerical semigroup $S$ is an $\\mathbf{I}(\\Delta)$-{\\it semigroup} if $S\\backslash \\{0\\}$ is an ideal of $\\Delta$. We will denote by $\\mathcal{J}(\\Delta)=\\{S \\mid S \\text{ is an $\\mathbf{I}(\\Delta)$-semigroup} \\}.$ We will say that $\\Delta$ is {\\it an ideal extension of } $S$ if $S\\in \\mathcal{J}(\\Delta).$ In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by $\\mathcal{J}^0(\\mathbb{N})=\\mathbb{N},$ $\\mathcal{J}^1(\\mathbb{N})=\\mathcal{J}(\\mathbb{N})$ and $\\mathcal{J}^{k+1}(\\mathbb{N})=\\mathcal{J}(\\mathcal{J}^{k}(\\mathbb{N}))$ for all $k\\in \\mathbb{N}.$ The complexity of a numerical semigroup $S$ is the minimun of the set $\\{k\\in \\mathbb{N}\\mid S \\in \\mathcal{J}^k(\\mathbb{N})\\}.$ In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.","url_abs":"https://arxiv.org/abs/2202.00920v1","url_pdf":"https://arxiv.org/pdf/2202.00920v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-complexity-of-a-numerical-semigroup","repo_url":"https://github.com/D-marina/CommutativeMonoids","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}