{"title":"On k-ary parts of maximal clones","authors":"","doi":"10.1007/s00012-024-00851-x","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the possible directions is to examine <em>k</em>-ary parts of the clones and their mutual inclusions. In this paper we study <em>k</em>-ary parts of maximal clones, for <span> <span>\\(k\\geqslant 2,\\)</span> </span> building on the already known results for their unary parts. It turns out that the poset of <em>k</em>-ary parts of maximal clones defined by central relations contains long chains.</p>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-03-18","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/s00012-024-00851-x","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
The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the possible directions is to examine k-ary parts of the clones and their mutual inclusions. In this paper we study k-ary parts of maximal clones, for \(k\geqslant 2,\) building on the already known results for their unary parts. It turns out that the poset of k-ary parts of maximal clones defined by central relations contains long chains.
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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