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

Abstract

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

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