{"title":"幂等生成数","authors":"N. A. KOLEGOV","doi":"10.1017/s0004972724000078","DOIUrl":null,"url":null,"abstract":"<p>The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304124645615-0035:S0004972724000078:S0004972724000078_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\lceil \\log _2 n\\rceil $</span></span></img></span></span> or <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304124645615-0035:S0004972724000078:S0004972724000078_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\lceil \\log _2 n\\rceil +1$</span></span></img></span></span>, where <span>n</span> is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"9 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"IDEMPOTENT GENERATORS OF INCIDENCE ALGEBRAS\",\"authors\":\"N. A. KOLEGOV\",\"doi\":\"10.1017/s0004972724000078\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304124645615-0035:S0004972724000078:S0004972724000078_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\lceil \\\\log _2 n\\\\rceil $</span></span></img></span></span> or <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304124645615-0035:S0004972724000078:S0004972724000078_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\lceil \\\\log _2 n\\\\rceil +1$</span></span></img></span></span>, where <span>n</span> is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.</p>\",\"PeriodicalId\":50720,\"journal\":{\"name\":\"Bulletin of the Australian Mathematical Society\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-03-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the Australian Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s0004972724000078\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0004972724000078","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either $\lceil \log _2 n\rceil $ or $\lceil \log _2 n\rceil +1$, where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.
期刊介绍:
Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way.
Published Bi-monthly
Published for the Australian Mathematical Society