On the semigroup BωF which is generated by the family F of atomic subsets of ω

Анотація

We study the semigroup ${\mathit{B}}_{\mathbf{\omega }}^{\mathcal{F}}$, which is introduced in [O. Gutik and M. Mykhalenych, On some generalization of the bicyclic monoid, Visnyk Lviv. Univ. Ser. Mech.-Mat. 90 (2020), 5-19], in the case when the family $\mathcal{F}$ of subsets of cardinality $\le 1$ in $\omega$. We show that ${\mathit{B}}_{\mathrm{\omega }}^{\mathcal{F}}$ is isomorphic to the  subsemigroup ${\mathcal{B}}_{\omega }^{\mapsto }\left({\mathit{F}}_{\mathbf{m}\mathbf{i}\mathbf{n}}\right)$ of the Brandt $\omega$-extension of the semilattice ${\mathit{F}}_{min}$ and describe all shift-continuous feebly compact $T_1$-topologies on the semigroup ${\mathcal{B}}_{\omega }^{\mapsto }\left({\mathit{F}}_{\mathbf{m}\mathbf{i}\mathbf{n}}\right)$. In particulary we prove that every shift-continuous feebly compact $T_1$-topology $\tau$ on ${\mathcal{B}}_{\omega }^{\mapsto }\left({\mathit{F}}_{\mathbf{m}\mathbf{i}\mathbf{n}}\right)$ is compact and moreover in this case  the space $\left({\mathcal{B}}_{\omega }^{\mapsto }\right({\mathit{F}}_{\mathbf{m}\mathbf{i}\mathbf{n}}),\tau )$ is homeomorphic to the one-point Alexandroff compactification of the discrete countable space ${\mathfrak{D}}_{\omega }$. We study the closure of ${\mathit{B}}_{\mathrm{\omega }}^{\mathcal{F}}$ in a semitopological semigroup. In particularly we show that ${\mathit{B}}_{\mathrm{\omega }}^{\mathcal{F}}$ is algebraically complete in the class of Hausdorff semitopological inverse semigroups with continuous inversion, and a Hausdorff topological inverse semigroup ${\mathit{B}}_{\mathrm{\omega }}^{\mathcal{F}}$ is closed in any Hausdorff topological semigroup if and only if the band $\mathit{E}\mathbf{\left(}{\mathit{B}}_{\mathrm{\omega }}^{\mathcal{F}}\mathbf{\right)}$ is compact.