{"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/characterizing-affine-mathcal-c-semigroups","title":"Characterizing affine $\\mathcal{C}$-semigroups","arxiv_id":"1907.03276","date":"2019-07-07","proceeding":null,"authors":["J. D. Díaz-Ramírez","J. I. García-García","D. Marín-Aragón","A. Vigneron-Tenorio"],"abstract":"Let $\\mathcal C \\subset \\mathbb N^p$ be a finitely generated integer cone and $S\\subset \\mathcal C$ be an affine semigroup such that the real cones generated by $\\mathcal C$ and by $S$ are equal. The semigroup $S$ is called $\\mathcal C$-semigroup if $\\mathcal C\\setminus S$ is a finite set. In this paper, we characterize the $\\mathcal C$-semigroups from their minimal generating sets, and we give an algorithm to check if $S$ is a $\\mathcal C$-semigroup and to compute its set of gaps. We also study the embedding dimension of $\\mathcal C$-semigroups obtaining a lower bound for it, and introduce some families of $\\mathcal C$-semigroups whose embedding dimension reaches our bound. In the last section, we present a method to obtain a decomposition of a $\\mathcal C$-semigroup into irreducible $\\mathcal C$-semigroups.","url_abs":"https://arxiv.org/abs/1907.03276v2","url_pdf":"https://arxiv.org/pdf/1907.03276v2.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":"characterizing-affine-mathcal-c-semigroups","repo_url":"https://github.com/D-marina/CommutativeMonoids","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}