{"title":"反射数值半群","authors":"Caleb M. Shor","doi":"10.51286/albjm/1675941730","DOIUrl":null,"url":null,"abstract":"We define a reflective numerical semigroup of genus g as a numerical semigroup that has a certain reflective symmetry when viewed within ℤ as an array with g columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo g. In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.","PeriodicalId":484514,"journal":{"name":"Albanian journal of mathematics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"REFLECTIVE NUMERICAL SEMIGROUPS\",\"authors\":\"Caleb M. Shor\",\"doi\":\"10.51286/albjm/1675941730\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We define a reflective numerical semigroup of genus g as a numerical semigroup that has a certain reflective symmetry when viewed within ℤ as an array with g columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo g. In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.\",\"PeriodicalId\":484514,\"journal\":{\"name\":\"Albanian journal of mathematics\",\"volume\":\"42 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-06-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Albanian journal of mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.51286/albjm/1675941730\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Albanian journal of mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.51286/albjm/1675941730","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We define a reflective numerical semigroup of genus g as a numerical semigroup that has a certain reflective symmetry when viewed within ℤ as an array with g columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo g. In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.
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