On compact topologies on the semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set

We study topologization of the semigroup $\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}{}_n(L)$ of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set $(L,⩽)$. In particular we show that every $T_1$ left-topological (right-topological) semigroup $\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}{}_n(L)$ is a completely Hausdorff, Urysohn, totally separated, scattered space. We prove that on the semigroup $\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}{}_n(L)$ admits a unique Hausdorff countably compact (pseudocompact) shift-continuous topology which is compact, and the Bohr compactification of a Hausdorff topological semigroup $\mathcal{O}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathcal{I}\text{\hspace{0.17em}⁣}{}_n(L)$ is the trivial semigroup.