{"title":"可除阶与多集序的维数","authors":"Milan Haiman","doi":"10.1007/s11083-023-09653-7","DOIUrl":null,"url":null,"abstract":"<p>The Dushnik–Miller dimension of a poset <i>P</i> is the least <i>d</i> for which <i>P</i> can be embedded into a product of <i>d</i> chains. Lewis and Souza isibility order on the interval of integers <span>\\([N/\\kappa , N]\\)</span> is bounded above by <span>\\(\\kappa (\\log \\kappa )^{1+o(1)}\\)</span> and below by <span>\\(\\Omega ((\\log \\kappa /\\log \\log \\kappa )^2)\\)</span>. We improve the upper bound to <span>\\(O((\\log \\kappa )^3/(\\log \\log \\kappa )^2).\\)</span> We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The Dimension of Divisibility Orders and Multiset Posets\",\"authors\":\"Milan Haiman\",\"doi\":\"10.1007/s11083-023-09653-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Dushnik–Miller dimension of a poset <i>P</i> is the least <i>d</i> for which <i>P</i> can be embedded into a product of <i>d</i> chains. Lewis and Souza isibility order on the interval of integers <span>\\\\([N/\\\\kappa , N]\\\\)</span> is bounded above by <span>\\\\(\\\\kappa (\\\\log \\\\kappa )^{1+o(1)}\\\\)</span> and below by <span>\\\\(\\\\Omega ((\\\\log \\\\kappa /\\\\log \\\\log \\\\kappa )^2)\\\\)</span>. We improve the upper bound to <span>\\\\(O((\\\\log \\\\kappa )^3/(\\\\log \\\\log \\\\kappa )^2).\\\\)</span> We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.</p>\",\"PeriodicalId\":501237,\"journal\":{\"name\":\"Order\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-11-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Order\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s11083-023-09653-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Order","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11083-023-09653-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The Dimension of Divisibility Orders and Multiset Posets
The Dushnik–Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza isibility order on the interval of integers \([N/\kappa , N]\) is bounded above by \(\kappa (\log \kappa )^{1+o(1)}\) and below by \(\Omega ((\log \kappa /\log \log \kappa )^2)\). We improve the upper bound to \(O((\log \kappa )^3/(\log \log \kappa )^2).\) We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.