On the semigroup $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$, which is generated by the family $\mathscr{F}_n$ of finite bounded intervals of $ω$ - Gutik, O.V.; Popadiuk, O.B. (Carpathian Mathematical Publications, 2023)
We study the semigroup $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$, which is introduced in the paper [Visnyk Lviv Univ. Ser. Mech.-Mat. 2020, 90, 5-19 (in Ukrainian)], in the case when the ${ω}$-closed family $\mathscr{F}_n$ generated by the set $\{0,1,…,n\}$. We show that the Green relations $\mathscr{D}$ and $\mathscr{J}$ coincide in $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$, the semigroup $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$ is isomorphic to the semigroup $\mathscr{I}_ω^{n+1}(\overrightarrow{conv})$ of partial convex order isomorphisms of $(ω,\leqslant)$ of the rank $\leqslant n+1$, and $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$ admits only Rees congruences. Also, we study shift-continuous topologies on the semigroup $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$. In particular, we prove that for any shift-continuous $T_1$-topology $τ$ on the semigroup $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$ every non-zero element of $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$ is an isolated point of $(\boldsymbol{B}_{ω}^{\mathscr{F}_n},τ)$, $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$ admits the unique compact shift-continuous $T_1$-topology, and every $ω_{\mathfrak{d}}$-compact shift-continuous $T_1$-topology is compact. We describe the closure of the semigroup $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$ in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup $\boldsymbol{B}_{ω}^{\mathscr{F}_n}$ is $H$-closed in the class of Hausdorff topological semigroups.