{"title":"关于双环半群和扩展双环半群的自同态","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":"{\"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}","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}
On endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup
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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