{"title":"Minimal Weak Drazin Inverses in Semigroups and Rings","authors":"Shuxian Xu, Jianlong Chen, Cang Wu","doi":"10.1007/s00009-024-02661-w","DOIUrl":null,"url":null,"abstract":"<p>In 1978, Campbell and Meyer proposed the notion of minimal rank weak Drazin inverses of complex matrices. In this paper, we define minimal weak Drazin inverses of elements in semigroups using Green’s preorder <span>\\(\\leqslant _{{\\mathcal {R}}},\\)</span> which generalize minimal rank weak Drazin inverses of complex matrices. For two elements <i>a</i>, <i>y</i> of a semigroup, it is proved that <i>y</i> is a minimal weak Drazin inverse of <i>a</i> if and only if <span>\\(ya^{k+1}=a^{k}\\)</span> for some nonnegative integer <i>k</i> and <span>\\(ay^{2}=y.\\)</span></p>","PeriodicalId":49829,"journal":{"name":"Mediterranean Journal of Mathematics","volume":"76 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mediterranean Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02661-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In 1978, Campbell and Meyer proposed the notion of minimal rank weak Drazin inverses of complex matrices. In this paper, we define minimal weak Drazin inverses of elements in semigroups using Green’s preorder \(\leqslant _{{\mathcal {R}}},\) which generalize minimal rank weak Drazin inverses of complex matrices. For two elements a, y of a semigroup, it is proved that y is a minimal weak Drazin inverse of a if and only if \(ya^{k+1}=a^{k}\) for some nonnegative integer k and \(ay^{2}=y.\)
期刊介绍:
The Mediterranean Journal of Mathematics (MedJM) is a publication issued by the Department of Mathematics of the University of Bari. The new journal replaces the Conferenze del Seminario di Matematica dell’Università di Bari which has been in publication from 1954 until 2003.
The Mediterranean Journal of Mathematics aims to publish original and high-quality peer-reviewed papers containing significant results across all fields of mathematics. The submitted papers should be of medium length (not to exceed 20 printed pages), well-written and appealing to a broad mathematical audience.
In particular, the Mediterranean Journal of Mathematics intends to offer mathematicians from the Mediterranean countries a particular opportunity to circulate the results of their researches in a common journal. Through such a new cultural and scientific stimulus the journal aims to contribute to further integration amongst Mediterranean universities, though it is open to contribution from mathematicians across the world.