{"title":"嵌入维数三的最长和最短分解","authors":"Baian Liu, JiaYan Yap","doi":"10.2140/involve.2023.16.673","DOIUrl":null,"url":null,"abstract":"For a numerical monoid $\\langle n_1, \\dots, n_k \\rangle$ minimally generated by $n_1, \\dots, n_k \\in \\mathbb{N}$ with $n_1<\\cdots<n_k$, the longest and shortest factorization lengths of an element $x$, denoted as $L(x)$ and $\\ell(x)$, respectively, follow the identities $L(x+n_1) = L(x) + 1$ and $\\ell(x+n_k) = \\ell(x) + 1$ for sufficiently large elements $x$. We characterize when these identities hold for all elements of numerical monoids of embedding dimension three.","PeriodicalId":36396,"journal":{"name":"Involve","volume":"74 ","pages":"0"},"PeriodicalIF":0.2000,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Longest and shortest factorizations in embedding dimension three\",\"authors\":\"Baian Liu, JiaYan Yap\",\"doi\":\"10.2140/involve.2023.16.673\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a numerical monoid $\\\\langle n_1, \\\\dots, n_k \\\\rangle$ minimally generated by $n_1, \\\\dots, n_k \\\\in \\\\mathbb{N}$ with $n_1<\\\\cdots<n_k$, the longest and shortest factorization lengths of an element $x$, denoted as $L(x)$ and $\\\\ell(x)$, respectively, follow the identities $L(x+n_1) = L(x) + 1$ and $\\\\ell(x+n_k) = \\\\ell(x) + 1$ for sufficiently large elements $x$. We characterize when these identities hold for all elements of numerical monoids of embedding dimension three.\",\"PeriodicalId\":36396,\"journal\":{\"name\":\"Involve\",\"volume\":\"74 \",\"pages\":\"0\"},\"PeriodicalIF\":0.2000,\"publicationDate\":\"2023-10-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Involve\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/involve.2023.16.673\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Involve","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/involve.2023.16.673","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}