{"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}
引用次数: 0
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 $\boldsymbol{B}_{\omega}^{\mathscr{F}}$.