{"title":"Acts with identities in the congruence lattice","authors":"I. B. Kozhuhov, A. M. Pryanichnikov","doi":"10.1007/s00012-022-00773-6","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that for any act <i>X</i> over a finite semigroup <i>S</i>, the congruence lattice <span>\\({{\\,\\mathrm{Con}\\,}}X\\)</span> embeds the lattice <span>\\({{\\,\\mathrm{Eq}\\,}}M\\)</span> of all equivalences of an infinite set <i>M</i> if and only if <i>X</i> is infinite. Equivalently: for an act <i>X</i> over a finite semigroup <i>S</i>, the lattice <span>\\({{\\,\\mathrm{Con}\\,}}X\\)</span> satisfies a non-trivial identity if and only if <i>X</i> is finite. Similar statements are proved for an act with zero over a completely 0-simple semigroup <span>\\({\\mathcal {M}}^0(G,I,\\Lambda ,P)\\)</span> where <span>\\(|G|,|I| <\\infty \\)</span>. We construct examples that show that the assumption <span>\\(|G|,|I| <\\infty \\)</span> is essential.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2022-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00012-022-00773-6","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: 1
Abstract
We prove that for any act X over a finite semigroup S, the congruence lattice \({{\,\mathrm{Con}\,}}X\) embeds the lattice \({{\,\mathrm{Eq}\,}}M\) of all equivalences of an infinite set M if and only if X is infinite. Equivalently: for an act X over a finite semigroup S, the lattice \({{\,\mathrm{Con}\,}}X\) satisfies a non-trivial identity if and only if X is finite. Similar statements are proved for an act with zero over a completely 0-simple semigroup \({\mathcal {M}}^0(G,I,\Lambda ,P)\) where \(|G|,|I| <\infty \). We construct examples that show that the assumption \(|G|,|I| <\infty \) is essential.
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.