On the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ which is generated by the family $\mathscr{F}$ of atomic subsets of $\omega$

Abstract

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.