{"title":"On the representation of ordered semigroups by transformations of ordered sets","authors":"Michael Tsingelis","doi":"10.1007/s00012-023-00810-y","DOIUrl":null,"url":null,"abstract":"<div><p>A transformation of an ordered set <i>M</i> is an isotone mapping of <i>M</i> into <i>M</i>. By a representation of an ordered semigroup <i>S</i> by transformations of an ordered set <i>M</i> we mean a homomorphism of <i>S</i> into the set of transformations of <i>M</i>, i.e. (since the set of transformations of <i>M</i> is an ordered semigroup) an isotone mapping from <i>S</i> into the set of transformations of <i>M</i> preserving the operations. We prove that this type of representation leads to an “action” of <i>S</i> on <i>M</i> and so we introduce the notion of a left operand of <i>M</i> over <i>S</i>. Also we introduce the notions of a left operator pseudoorder on a left operand over <i>S</i> and a left operator homomorphism between left operands over <i>S</i>. We show that the concept of left operator pseudoorders on left operands over <i>S</i> plays an important role in the study of left operator homomorphisms of left operands over <i>S</i>. In the case of right operands over <i>S</i> dually definitions and results hold.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00810-y.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00012-023-00810-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
A transformation of an ordered set M is an isotone mapping of M into M. By a representation of an ordered semigroup S by transformations of an ordered set M we mean a homomorphism of S into the set of transformations of M, i.e. (since the set of transformations of M is an ordered semigroup) an isotone mapping from S into the set of transformations of M preserving the operations. We prove that this type of representation leads to an “action” of S on M and so we introduce the notion of a left operand of M over S. Also we introduce the notions of a left operator pseudoorder on a left operand over S and a left operator homomorphism between left operands over S. We show that the concept of left operator pseudoorders on left operands over S plays an important role in the study of left operator homomorphisms of left operands over S. In the case of right operands over S dually definitions and results hold.
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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