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Let $\boldsymbol{B}_{[0,\infty)}$ be the semigroup which is defined in theAhre paper \cite{Ahre=1981}. The semigroup $\boldsymbol{B}_{[0,\infty)}$ withthe induced usual topology $\tau_u$ from $\mathbb{R}^2$, with the topology$\tau_L$ which is generated by the natural partial order on$\boldsymbol{B}_{[0,\infty)}$, 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 anadjoined 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 thatif $S_{\mathfrak{d}}^I$ is a Hausdorff locally compact semitopologicalsemigroup $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$ with an adjoined compactideal $I$ then $I$ is an open subset of $S_{\mathfrak{d}}^I$.

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