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

Abstract

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

13 pages

Cited by in corpus (1)

• On the semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ which is generated by the family ${\mathcal{F}}_n$ of finite bounded intervals of $\omega$