On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal

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 $S_{\mathfrak{d}}^I$.