  We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and
$\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the
following properties: every Hausdorff left-continuous (right-continuous)
topology on $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) is discrete and
there exists a compact Hausdorff topological monoid $S$ which contains
$\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) as a submonoid. Also, we
construct a non-discrete right-continuous (left-continuous) topology $\tau_p^+$
($\tau_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$)
which is not left-continuous (right-continuous).
