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

Let ${\mathit{B}}_{[0,\mathrm{\infty })}$ be the semigroup which is defined in the Ahre paper \cite{Ahre=1981}. The semigroup ${\mathit{B}}_{[0,\mathrm{\infty })}$ 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 ${\mathit{B}}_{[0,\mathrm{\infty })}$, and the discrete topology are denoted by ${\mathit{B}}_{[0,\mathrm{\infty })}^1$, ${\mathit{B}}_{[0,\mathrm{\infty })}^2$, and ${\mathit{B}}_{[0,\mathrm{\infty })}^{\mathfrak{d}}$, respectively. We show that if $S_1^I$ ($S_2^I$) is a Hausdorff locally compact semitopological semigroup ${\mathit{B}}_{[0,\mathrm{\infty })}^1$ (${\mathit{B}}_{[0,\mathrm{\infty })}^2$) with an adjoined compact ideal $I$ then either $I$ is an open subset of $S_1^I$ ($S_2^I$) or the semigroup $S_1^I$ ($S_2^I$) is compact. Also, we proved that if $S_{\mathfrak{d}}^I$ is a Hausdorff locally compact semitopological semigroup ${\mathit{B}}_{[0,\mathrm{\infty })}^{\mathfrak{d}}$ with an adjoined compact ideal $I$ then $I$ is an open subset of $S_{\mathfrak{d}}^I$.