{"title":"有限几何格的秩不等式","authors":"Curtis Greene","doi":"10.1016/S0021-9800(70)80090-4","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>L</em> be a finite geometric lattice of dimension <em>n</em>, and let <em>w(k)</em> denote the number of elements in <em>L</em> of rank <em>k</em>. Two theorems about the numbers <em>w(k)</em> are proved: first, <em>w(k)</em>≥<em>w</em>(1) for <em>k</em>=2,3,…,<em>n</em>−1. Second, <em>w(k)</em>=<em>w</em>(1) if and only if <em>k</em>=<em>n</em>−1 and <em>L</em> is modular. Several corollaries concerning the “matching” of points and dual points are derived from these results.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 4","pages":"Pages 357-364"},"PeriodicalIF":0.0000,"publicationDate":"1970-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80090-4","citationCount":"54","resultStr":"{\"title\":\"A rank inequality for finite geometric lattices\",\"authors\":\"Curtis Greene\",\"doi\":\"10.1016/S0021-9800(70)80090-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <em>L</em> be a finite geometric lattice of dimension <em>n</em>, and let <em>w(k)</em> denote the number of elements in <em>L</em> of rank <em>k</em>. Two theorems about the numbers <em>w(k)</em> are proved: first, <em>w(k)</em>≥<em>w</em>(1) for <em>k</em>=2,3,…,<em>n</em>−1. Second, <em>w(k)</em>=<em>w</em>(1) if and only if <em>k</em>=<em>n</em>−1 and <em>L</em> is modular. Several corollaries concerning the “matching” of points and dual points are derived from these results.</p></div>\",\"PeriodicalId\":100765,\"journal\":{\"name\":\"Journal of Combinatorial Theory\",\"volume\":\"9 4\",\"pages\":\"Pages 357-364\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1970-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80090-4\",\"citationCount\":\"54\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021980070800904\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021980070800904","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k)≥w(1) for k=2,3,…,n−1. Second, w(k)=w(1) if and only if k=n−1 and L is modular. Several corollaries concerning the “matching” of points and dual points are derived from these results.
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