{"title":"Thick sets are exactly the sets with Følner density 1","authors":"Neil Hindman, Dona Strauss","doi":"10.1007/s00233-024-10456-4","DOIUrl":null,"url":null,"abstract":"<p><i>Følner density</i> is a very natural notion of density which is defined on any semigroup satisfying the Strong Følner Condition (SFC). (These include all commutative semigroups and all left cancellative left amenable semigroups.) <i>Piecewise syndetic</i> and <i>thick</i> are notions of largeness arising from topological dynamics. It has been known that if <i>S</i> satisfies SFC and is either left cancellative or satisfies a weak right cancellation requirement, then every thick subset has density 1. We show here that in any semigroup <i>S</i> satisfying SFC a subset of <i>S</i> is thick if and only if it has density 1. As a consequence, every piecewise syndetic set has positive density.</p>","PeriodicalId":49549,"journal":{"name":"Semigroup Forum","volume":"61 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Semigroup Forum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00233-024-10456-4","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Følner density is a very natural notion of density which is defined on any semigroup satisfying the Strong Følner Condition (SFC). (These include all commutative semigroups and all left cancellative left amenable semigroups.) Piecewise syndetic and thick are notions of largeness arising from topological dynamics. It has been known that if S satisfies SFC and is either left cancellative or satisfies a weak right cancellation requirement, then every thick subset has density 1. We show here that in any semigroup S satisfying SFC a subset of S is thick if and only if it has density 1. As a consequence, every piecewise syndetic set has positive density.
福尔纳密度是一个非常自然的密度概念，它定义在任何满足强福尔纳条件（SFC）的半群上（包括所有交换半群和所有左可抵消半群）。片状联合和厚是产生于拓扑动力学的大型概念。众所周知，如果 S 满足 SFC，并且是左可消的或满足弱右可消的要求，那么每个厚子集的密度都是 1。我们在此证明，在任何满足 SFC 的半群 S 中，当且仅当 S 的一个子集具有密度 1 时，它就是厚子集。因此，每个片断联合集都有正密度。
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