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

Abstract

We study the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$, which is introduced in the paper [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 Ukrainian)], in the case when the family ${\mathcal{F}}_n$ generated by the set $\{0,1,\dots ,n\}$. We show that the Green relations $\mathcal{D}$ and $\mathcal{J}$ coincide in ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$, the semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ is isomorphic to the semigroup ${\mathcal{I}}_{\omega }^{n+1}(\overrightarrow{\mathrm{conv}})$ of partial convex order isomorphisms of $(\omega ,⩽)$ of the rank $⩽n+1$, and ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ admits only Rees congruences. Also, we study shift-continuous topologies on the semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$. In particular we prove that for any shift-continuous $T_1$-topology $\tau$ on the semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ every non-zero element of ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ is an isolated point of $({\mathbf{B}}_{\omega }^{{\mathcal{F}}_n},\tau )$, ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ admits the unique compact shift-continuous $T_1$-topology, and every ${\omega }_{\mathfrak{d}}$-compact shift-continuous $T_1$-topology is compact. We describe the closure of the semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ is $H$-closed in the class of Hausdorff topological semigroups.

19 pages, 1 figure

References in corpus (3)

• Semigroup Closures of Finite Rank Symmetric Inverse Semigroups
• On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise
• On the semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ which is generated by the family $\mathcal{F}$ of atomic subsets of $\omega$

Cited by in corpus (1)

• The semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set