{"title":"关于半拓扑半群$\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$当一个族$\\mathscr{F}$由$\\omega的归纳非空子集组成时$","authors":"O. Gutik, M. Mykhalenych","doi":"10.30970/ms.59.1.20-28","DOIUrl":null,"url":null,"abstract":"Let $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$ be the bicyclic semigroup extension for the family $\\mathscr{F}$ of ${\\omega}$-closed subsets of $\\omega$ which is introduced in \\cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$ for the family $\\mathscr{F}$ of inductive ${\\omega}$-closed subsets of $\\omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \\cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$ as a proper dense subsemigroup then $S\\setminus\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}\\sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $\\boldsymbol{B}_{\\omega}^{\\mathscr{F}}$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On a semitopological semigroup $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ when a family $\\\\mathscr{F}$ consists of inductive non-empty subsets of $\\\\omega$\",\"authors\":\"O. Gutik, M. Mykhalenych\",\"doi\":\"10.30970/ms.59.1.20-28\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ be the bicyclic semigroup extension for the family $\\\\mathscr{F}$ of ${\\\\omega}$-closed subsets of $\\\\omega$ which is introduced in \\\\cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ for the family $\\\\mathscr{F}$ of inductive ${\\\\omega}$-closed subsets of $\\\\omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \\\\cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ as a proper dense subsemigroup then $S\\\\setminus\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}\\\\sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $\\\\boldsymbol{B}_{\\\\omega}^{\\\\mathscr{F}}$.\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-12-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Matematychni Studii\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30970/ms.59.1.20-28\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.59.1.20-28","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
On a semitopological semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$
Let $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ be the bicyclic semigroup extension for the family $\mathscr{F}$ of ${\omega}$-closed subsets of $\omega$ which is introduced in \cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ for the family $\mathscr{F}$ of inductive ${\omega}$-closed subsets of $\omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ as a proper dense subsemigroup then $S\setminus\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}\sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$.