{"title":"Lattices of retracts of direct products of two finite chains and notes on retracts of lattices","authors":"Gábor Czédli","doi":"10.1007/s00012-022-00788-z","DOIUrl":null,"url":null,"abstract":"<div><p>Ordered by set inclusion, the <i>retracts</i> of a lattice <i>L</i> together with the empty set form a bounded poset <span>\\(Ret (L)\\)</span>. By a <i>grid</i> we mean the direct product of two non-singleton finite chains. We prove that if <i>G</i> is a grid, then <span>\\(Ret (G)\\)</span> is a lattice. We determine the number of elements of <span>\\(Ret (G)\\)</span>. Some easy properties of retracts, <i>retractions</i>, and their kernels called <i>retraction congruences</i> of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice <i>M</i> such that <span>\\(Ret (M)\\)</span> is not a lattice.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2022-08-02","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/s00012-022-00788-z","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
Ordered by set inclusion, the retracts of a lattice L together with the empty set form a bounded poset \(Ret (L)\). By a grid we mean the direct product of two non-singleton finite chains. We prove that if G is a grid, then \(Ret (G)\) is a lattice. We determine the number of elements of \(Ret (G)\). Some easy properties of retracts, retractions, and their kernels called retraction congruences of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice M such that \(Ret (M)\) is not a lattice.
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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