{"title":"Semilattices of stratified extensions","authors":"James Renshaw, William Warhurst","doi":"10.1007/s00233-024-10447-5","DOIUrl":null,"url":null,"abstract":"<p>Grillet (Semigroup Forum 50:25–36, 1995) introduced the concept of stratified semigroups as a kind of generalisation of finite nilsemigroups. We extend Grillet’s ideas by introducing the notion of the base of a semigroup and show that a semigroup is stratified if and only if its base is either empty or consists of only the zero element. The general structure of semigroups with non-trivial bases is studied and we show that these can be described in terms of ideal extensions of semigroups by stratified semigroups. We consider certain types of group-bound semigroups and also ideal extensions of Clifford semigroups, and show how to describe them as semilattices of ideal extensions by stratified semigroups and provide a number of interesting examples.</p>","PeriodicalId":49549,"journal":{"name":"Semigroup Forum","volume":"2015 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-06-24","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/s00233-024-10447-5","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
Grillet (Semigroup Forum 50:25–36, 1995) introduced the concept of stratified semigroups as a kind of generalisation of finite nilsemigroups. We extend Grillet’s ideas by introducing the notion of the base of a semigroup and show that a semigroup is stratified if and only if its base is either empty or consists of only the zero element. The general structure of semigroups with non-trivial bases is studied and we show that these can be described in terms of ideal extensions of semigroups by stratified semigroups. We consider certain types of group-bound semigroups and also ideal extensions of Clifford semigroups, and show how to describe them as semilattices of ideal extensions by stratified semigroups and provide a number of interesting examples.
格里列特（Semigroup Forum 50:25-36, 1995）提出了分层半群的概念，作为有限无半群的一种概括。我们通过引入半群基的概念扩展了格里列特的观点，并证明当且仅当一个半群的基为空或仅由零元素组成时，该半群是分层的。我们研究了具有非琐基的半群的一般结构，并证明这些半群可以用分层半群的理想扩展来描述。我们考虑了某些类型的群约束半群以及克利福德半群的理想扩展，并说明了如何将它们描述为分层半群理想扩展的半网格，还提供了一些有趣的例子。
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Semigroup Forum is a platform for speedy and efficient transmission of information on current research in semigroup theory.
Scope: Algebraic semigroups, topological semigroups, partially ordered semigroups, semigroups of measures and harmonic analysis on semigroups, numerical semigroups, transformation semigroups, semigroups of operators, and applications of semigroup theory to other disciplines such as ring theory, category theory, automata, logic, etc.
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