  We study the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$, which is
introduced in the paper [O. Gutik and M. Mykhalenych, \emph{On some
generalization of the bicyclic monoid}, Visnyk Lviv. Univ. Ser. Mech.-Mat.
\textbf{90} (2020), 5--19 (in Ukrainian)], in the case when the family
$\mathscr{F}_n$ generated by the set $\{0,1,\ldots,n\}$. We show that the Green
relations $\mathscr{D}$ and $\mathscr{J}$ coincide in
$\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$, the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is isomorphic to the semigroup
$\mathscr{I}_\omega^{n+1}(\overrightarrow{\mathrm{conv}})$ of partial convex
order isomorphisms of $(\omega,\leqslant)$ of the rank $\leqslant n+1$, and
$\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ admits only Rees congruences. Also,
we study shift-continuous topologies on the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$. In particular we prove that for any
shift-continuous $T_1$-topology $\tau$ on the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ every non-zero element of
$\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is an isolated point of
$(\boldsymbol{B}_{\omega}^{\mathscr{F}_n},\tau)$,
$\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ admits the unique compact
shift-continuous $T_1$-topology, and every $\omega_{\mathfrak{d}}$-compact
shift-continuous $T_1$-topology is compact. We describe the closure of the
semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ in a Hausdorff
semitopological semigroup and prove the criterium when a topological inverse
semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is $H$-closed in the class
of Hausdorff topological semigroups.
