{"title":"零半群中的高交换子","authors":"Jelena Radović, Nebojša Mudrinski","doi":"10.1007/s00012-023-00809-5","DOIUrl":null,"url":null,"abstract":"<div><p>We introduce the notion of the higher commutator of ideals in semigroups. For semigroups with zero, it is shown that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero all four conditions of solvability, supernilpotency, nilpotency and nilpotency in the sense of semigroup theory, are equivalent.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00809-5.pdf","citationCount":"0","resultStr":"{\"title\":\"Higher commutators in semigroups with zero\",\"authors\":\"Jelena Radović, Nebojša Mudrinski\",\"doi\":\"10.1007/s00012-023-00809-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We introduce the notion of the higher commutator of ideals in semigroups. For semigroups with zero, it is shown that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero all four conditions of solvability, supernilpotency, nilpotency and nilpotency in the sense of semigroup theory, are equivalent.</p></div>\",\"PeriodicalId\":50827,\"journal\":{\"name\":\"Algebra Universalis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00012-023-00809-5.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra Universalis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00012-023-00809-5\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00012-023-00809-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
We introduce the notion of the higher commutator of ideals in semigroups. For semigroups with zero, it is shown that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero all four conditions of solvability, supernilpotency, nilpotency and nilpotency in the sense of semigroup theory, are equivalent.
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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