On a semitopological semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ when a family $\mathcal{F}$ consists of inductive non-empty subsets of $\omega$

Abstract

Let ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ be the bicyclic semigroup extension for the family $\mathcal{F}$ of $\omega$-closed subsets of $\omega$ which is introduced in \cite{Gutik-Mykhalenych=2020}. We study topologizations of the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ for the family $\mathcal{F}$ of inductive $\omega$-closed subsets of $\omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ as a proper dense subsemigroup then $S\setminus {\mathbf{B}}_{\omega }^{\mathcal{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}\bigsqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$.

8 pages. arXiv admin note: substantial text overlap with arXiv:2112.15000

References in corpus (1)

• On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero