  In the paper it is shown that every Hausdorff locally compact semigroup
topology on the extended bicyclic semigroup with adjoined zero
$\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ is discrete, but on
$\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ there exist $\mathfrak{c}$ many
different Hausdorff locally compact shift-continuous topologies. Also, it is
constructed on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ the unique minimal shift
continuous topology and the unique minimal inverse semigroup topology.
