{"title":"Relative lengths of Maltsev conditions","authors":"Paolo Lipparini","doi":"10.1007/s00012-024-00855-7","DOIUrl":null,"url":null,"abstract":"<div><p>We solve some problems about relative lengths of Maltsev conditions, in particular, we provide an affirmative answer to a classical problem raised by A. Day more than 50 years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms <span>\\(t_0,\\ldots , t_n\\)</span> witnessing congruence distributivity it is possible to construct terms <span>\\(u_0,\\ldots , u _{2n-1} \\)</span> witnessing congruence modularity. We show that Day’s result about the number of such terms is sharp when <i>n</i> is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, as well as with possible variations we will call “specular” and “defective”. All the results hold when restricted to locally finite varieties.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00855-7.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00012-024-00855-7","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
We solve some problems about relative lengths of Maltsev conditions, in particular, we provide an affirmative answer to a classical problem raised by A. Day more than 50 years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms \(t_0,\ldots , t_n\) witnessing congruence distributivity it is possible to construct terms \(u_0,\ldots , u _{2n-1} \) witnessing congruence modularity. We show that Day’s result about the number of such terms is sharp when n is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, as well as with possible variations we will call “specular” and “defective”. All the results hold when restricted to locally finite varieties.
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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