We study the semigroup $\boldsymbol{ B}_{ \omega}^{ \mathscr{ F}}$, which is introduced in [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 the case when the family $\mathscr{ F}$ of subsets of cardinality $\leqslant 1$ in $\omega$. We show that $\boldsymbol{ B}_{ \omega}^{ \mathscr{ F}}$ is isomorphic to the subsemigroup $\mathscr{ B}_{ \omega}^{ \Rsh}(\boldsymbol{ F}_{ \min})$ of the Brandt $\omega$-extension of the semilattice $\boldsymbol{ F}_{ \min}$ and describe all shift-continuous feebly compact $T_1$-topologies on the semigroup $\mathscr{ B}_{ \omega}^{ \Rsh}(\boldsymbol{ F}_{ \min})$. In particulary we prove that every shift-continuous feebly compact $T_1$-topology $\tau$ on $\mathscr{ B}_{ \omega}^{ \Rsh}(\boldsymbol{ F}_{ \min})$ is compact and moreover in this case the space $(\mathscr{ B}_{ \omega}^{ \Rsh}(\boldsymbol{ F}_{ \min}),\tau)$ is homeomorphic to the one-point Alexandroff compactification of the discrete countable space $\mathfrak{ D}(\omega)$. We study the closure of $\boldsymbol{ B}_{ \omega}^{ \mathscr{ F}}$ in a semitopological semigroup. In particularly we show that $\boldsymbol{ B}_{ \omega}^{ \mathscr{ F}}$ is algebraically complete in the class of Hausdorff semitopological inverse semigroups with continuous inversion, and a Hausdorff topological inverse semigroup $\boldsymbol{ B}_{ \omega}^{ \mathscr{ F}}$ is closed in any Hausdorff topological semigroup if and only if the band $E(\boldsymbol{ B}_{ \omega}^{ \mathscr{ F}})$ is compact.
Categories: math.GR math.GN