{"title":"Strongly minimal Steiner systems II: coordinatization and quasigroups","authors":"John T. Baldwin","doi":"10.1007/s00012-023-00812-w","DOIUrl":null,"url":null,"abstract":"<div><p>Each strongly minimal Steiner <i>k</i>-system (<i>M</i>, <i>R</i>) (where is <i>R</i> is a ternary collinearity relation) can be ‘coordinatized’ in the sense of (Ganter–Werner 1975) by a quasigroup if <i>k</i> is a prime-power. We show this coordinatization is never definable in (<i>M</i>, <i>R</i>) and the strongly minimal Steiner <i>k</i>-systems constructed in (Baldwin–Paolini 2020) never interpret a quasigroup. Nevertheless, by refining the construction, if <i>k</i> is a prime power, in each (2, <i>k</i>)-variety of quasigroups (Definition 3.10) there is a strongly minimal quasigroup that interprets a Steiner <i>k</i>-system.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-05-02","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/s00012-023-00812-w","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
Each strongly minimal Steiner k-system (M, R) (where is R is a ternary collinearity relation) can be ‘coordinatized’ in the sense of (Ganter–Werner 1975) by a quasigroup if k is a prime-power. We show this coordinatization is never definable in (M, R) and the strongly minimal Steiner k-systems constructed in (Baldwin–Paolini 2020) never interpret a quasigroup. Nevertheless, by refining the construction, if k is a prime power, in each (2, k)-variety of quasigroups (Definition 3.10) there is a strongly minimal quasigroup that interprets a Steiner k-system.
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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