{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T22:41:05Z","timestamp":1785883265797,"version":"3.56.0"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,14]],"date-time":"2024-03-14T00:00:00Z","timestamp":1710374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, we will introduce the concept of ratio-covariety, as a family R of numerical semigroups that has a minimum, denoted by min(R), is closed under intersection, and if S\u2208R and S\u2260min(R), then S\\{r(S)}\u2208R, where r(S) denotes the ratio of S. The notion of ratio-covariety will allow us to: (1) describe an algorithmic procedure to compute R; (2) prove the existence of the smallest element of R that contains a set of positive integers; and (3) talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. In addition, in this paper we will apply the previous results to the study of the ratio-covariety R(F,m)={S\u2223S\u00a0is\u00a0a\u00a0numerical\u00a0semigroup\u00a0with\u00a0Frobenius\u00a0number\u00a0F\u00a0and\u00a0multiplicitym}.<\/jats:p>","DOI":"10.3390\/axioms13030193","type":"journal-article","created":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T04:47:05Z","timestamp":1710478025000},"page":"193","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Ratio-Covarieties of Numerical Semigroups"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0593-9434","authenticated-orcid":false,"given":"Mar\u00eda \u00c1ngeles","family":"Moreno-Fr\u00edas","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, University of C\u00e1diz, E-11510 Puerto Real, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jos\u00e9 Carlos","family":"Rosales","sequence":"additional","affiliation":[{"name":"Department of Algebra, Faculty of Sciences, University of Granada, E-18071 Granada, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Rosales, J.C., and Garc\u00eda-S\u00e1nchez, P.A. (2009). Numerical Semigroups, Springer.","DOI":"10.1007\/978-1-4419-0160-6"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.2307\/2373418","article-title":"Local rings of high embedding dimension","volume":"89","author":"Abhyankar","year":"1967","journal-title":"Am. J. Math."},{"key":"ref_3","first-page":"1","article-title":"Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analitycally Irreducible Local Domains","volume":"598","author":"Barucci","year":"1997","journal-title":"Memoirs Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/0021-8693(92)90118-6","article-title":"One dimensional local rings of maximal and almost maximal length","volume":"151","author":"Brown","year":"2008","journal-title":"J. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/0021-8693(79)90331-4","article-title":"Cohen-Macaualy local rings of maximal embedding dimension","volume":"56","author":"Sally","year":"1979","journal-title":"J. Algebra"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ram\u00edrez Alfons\u00edn, J.L. (2005). The Diophantine Frobenius Problem, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198568209.001.0001"},{"key":"ref_7","first-page":"142","article-title":"Mathematical question with their solutions","volume":"41","author":"Sylvester","year":"1984","journal-title":"Educ. Times"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/BF01300131","article-title":"Complexity of the Frobenius problem","volume":"16","year":"1996","journal-title":"Combinatorica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"147","DOI":"10.4171\/pm\/2064","article-title":"The set of numerical semigroups of a given multiplicity and Frobenius number","volume":"78","author":"Branco","year":"2021","journal-title":"Port. Math."},{"key":"ref_10","unstructured":"(2024, January 09). The GAP Group, GAP\u2014Groups, Algorithms, and Programming, Version 4.12.2. 2022. Available online: https:\/\/www.gapsystem.org."},{"key":"ref_11","unstructured":"Delgado, M., Garc\u00eda-S\u00e1nchez, P.A., and Morais, J. (2024, January 09). NumericalSgps, A Package for Numerical Semigroups, Version 1.3.1 (2022), (Refereed GAP Package). Available online: https:\/\/gap-packages.github.io\/numericalsgps."},{"key":"ref_12","first-page":"1198","article-title":"Sur les branches superlin\u00e9aires des courbes alg\u00e9briques","volume":"222","year":"1946","journal-title":"C.R. Acad. Sci. Paris"},{"key":"ref_13","first-page":"63","article-title":"On numerical semigroups","volume":"35","author":"Gottlieb","year":"1987","journal-title":"Semigroup Forum"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1090\/S0002-9939-1970-0265353-7","article-title":"The value-semigroup of a one-dimensional Gorenstein ring","volume":"25","author":"Kunz","year":"1973","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1142\/S0218196711006492","article-title":"Irreducibility in the set of numerical semigroups with fixed multiplicity","volume":"21","author":"Blanco","year":"2011","journal-title":"Int. J. Algebra Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/193\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:13:36Z","timestamp":1760105616000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/193"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,14]]},"references-count":15,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["axioms13030193"],"URL":"https:\/\/doi.org\/10.3390\/axioms13030193","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,14]]}}}