{"title":"On locally compact shift continuous topologies on the semigroup $\\boldsymbol{B}_{[0,\\infty)}$ with an adjoined compact ideal","authors":"O. Gutik, Markian Khylynskyi","doi":"10.30970/ms.61.1.10-21","DOIUrl":null,"url":null,"abstract":"Let $[0,\\infty)$ be the set of all non-negative real numbers. The set $\\boldsymbol{B}_{[0,\\infty)}=[0,\\infty)\\times [0,\\infty)$ with the following binary operation $(a,b)(c,d)=(a+c-\\min\\{b,c\\},b+d-\\min\\{b,c\\})$ is a bisimple inverse semigroup.In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup $\\boldsymbol{B}_{[0,\\infty)}$ with an adjoined compact ideal of the following tree types.The semigroup $\\boldsymbol{B}_{[0,\\infty)}$ with the induced usual topology $\\tau_u$ from $\\mathbb{R}^2$, with the topology $\\tau_L$ which is generated by the natural partial order on the inverse semigroup $\\boldsymbol{B}_{[0,\\infty)}$, and the discrete topology are denoted by $\\boldsymbol{B}^1_{[0,\\infty)}$, $\\boldsymbol{B}^2_{[0,\\infty)}$, and $\\boldsymbol{B}^{\\mathfrak{d}}_{[0,\\infty)}$, respectively. We show that if $S_1^I$ ($S_2^I$) is a Hausdorff locally compact semitopological semigroup $\\boldsymbol{B}^1_{[0,\\infty)}$ ($\\boldsymbol{B}^2_{[0,\\infty)}$) with an adjoined compact ideal $I$ then either $I$ is an open subset of $S_1^I$ ($S_2^I$) or the topological space $S_1^I$ ($S_2^I$) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on $S^1_{\\boldsymbol{0}}=\\boldsymbol{B}^1_{[0,\\infty)}\\cup\\{\\boldsymbol{0}\\}$ (resp. $S^2_{\\boldsymbol{0}}=\\boldsymbol{B}^2_{[0,\\infty)}\\cup\\{\\boldsymbol{0}\\}$) with an adjoined zero $\\boldsymbol{0}$ is either homeomorphic to the one-point Alexandroff compactification of the topological space $\\boldsymbol{B}^1_{[0,\\infty)}$ (resp. $\\boldsymbol{B}^2_{[0,\\infty)}$) or zero is an isolated point of $S^1_{\\boldsymbol{0}}$ (resp. $S^2_{\\boldsymbol{0}}$).Also, we proved that if $S_{\\mathfrak{d}}^I$ is a Hausdorff locally compact semitopological semigroup $\\boldsymbol{B}^{\\mathfrak{d}}_{[0,\\infty)}$ with an adjoined compact ideal $I$ then $I$ is an open subset of $S_{\\mathfrak{d}}^I$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-01-12","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.61.1.10-21","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 $[0,\infty)$ be the set of all non-negative real numbers. The set $\boldsymbol{B}_{[0,\infty)}=[0,\infty)\times [0,\infty)$ with the following binary operation $(a,b)(c,d)=(a+c-\min\{b,c\},b+d-\min\{b,c\})$ is a bisimple inverse semigroup.In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal of the following tree types.The semigroup $\boldsymbol{B}_{[0,\infty)}$ with the induced usual topology $\tau_u$ from $\mathbb{R}^2$, with the topology $\tau_L$ which is generated by the natural partial order on the inverse semigroup $\boldsymbol{B}_{[0,\infty)}$, and the discrete topology are denoted by $\boldsymbol{B}^1_{[0,\infty)}$, $\boldsymbol{B}^2_{[0,\infty)}$, and $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$, respectively. We show that if $S_1^I$ ($S_2^I$) is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^1_{[0,\infty)}$ ($\boldsymbol{B}^2_{[0,\infty)}$) with an adjoined compact ideal $I$ then either $I$ is an open subset of $S_1^I$ ($S_2^I$) or the topological space $S_1^I$ ($S_2^I$) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on $S^1_{\boldsymbol{0}}=\boldsymbol{B}^1_{[0,\infty)}\cup\{\boldsymbol{0}\}$ (resp. $S^2_{\boldsymbol{0}}=\boldsymbol{B}^2_{[0,\infty)}\cup\{\boldsymbol{0}\}$) with an adjoined zero $\boldsymbol{0}$ is either homeomorphic to the one-point Alexandroff compactification of the topological space $\boldsymbol{B}^1_{[0,\infty)}$ (resp. $\boldsymbol{B}^2_{[0,\infty)}$) or zero is an isolated point of $S^1_{\boldsymbol{0}}$ (resp. $S^2_{\boldsymbol{0}}$).Also, we proved that if $S_{\mathfrak{d}}^I$ is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$ with an adjoined compact ideal $I$ then $I$ is an open subset of $S_{\mathfrak{d}}^I$.