{"title":"Bubble lattices I: Structure","authors":"","doi":"10.1007/s00012-024-00842-y","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of this structure. We attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the <em>bubble lattice</em>. We characterize the bubble lattice both locally (via certain transformations of shuffle words) and globally (using a notion of inversion set). We then prove that the bubble lattice is extremal and constructable by interval doublings. Lastly, we prove that our bubble lattice is a generalization of the Hochschild lattice studied earlier by Chapoton, Combe and the second author.</p>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00012-024-00842-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of this structure. We attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the bubble lattice. We characterize the bubble lattice both locally (via certain transformations of shuffle words) and globally (using a notion of inversion set). We then prove that the bubble lattice is extremal and constructable by interval doublings. Lastly, we prove that our bubble lattice is a generalization of the Hochschild lattice studied earlier by Chapoton, Combe and the second author.
摘要 C. 格林引入了洗牌晶格作为 DNA 变异的理想化模型，并发现了这种结构的显著组合和枚举特性。我们试图从网格理论的角度来解释这些特性。为此，我们引入并研究了洗牌晶格的阶次扩展--气泡晶格。我们从局部（通过洗牌词的某些变换）和全局（使用反转集的概念）两方面描述了气泡网格的特征。然后，我们证明气泡网格是极值网格，可以通过区间倍增来构造。最后，我们证明我们的气泡网格是查波顿、康贝和第二作者早先研究的霍赫希尔德网格的广义化。
期刊介绍：
Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.
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