{"title":"On endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup","authors":"O. Gutik, Oksana Prokhorenkova, D. Sekh","doi":"10.30970/vmm.2021.92.005-016","DOIUrl":null,"url":null,"abstract":"It is proved that the semigroups $\\mathrm{\\mathbf{End}}(\\boldsymbol{B}_{\\omega})$ and $\\mathrm{\\mathbf{End}}(\\boldsymbol{B}_{\\mathbb{Z}})$ of the endomorphisms of the bicyclic semigroup $\\boldsymbol{B}_{\\omega}$ and the endomorphisms of the extended bicyclic semigroup $\\boldsymbol{B}_{\\mathbb{Z}}$ are isomorphic to the semidirect products $(\\omega,+)\\rtimes_\\varphi(\\omega,*)$ and $\\mathbb{Z}(+)\\rtimes_\\varphi(\\omega,*)$, respectively.","PeriodicalId":277870,"journal":{"name":"Visnyk Lvivskogo Universytetu Seriya Mekhaniko-Matematychna","volume":"62 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Visnyk Lvivskogo Universytetu Seriya Mekhaniko-Matematychna","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/vmm.2021.92.005-016","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 4
Abstract
It is proved that the semigroups $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\omega})$ and $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\mathbb{Z}})$ of the endomorphisms of the bicyclic semigroup $\boldsymbol{B}_{\omega}$ and the endomorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ are isomorphic to the semidirect products $(\omega,+)\rtimes_\varphi(\omega,*)$ and $\mathbb{Z}(+)\rtimes_\varphi(\omega,*)$, respectively.
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