Symmetric inverse topological semigroups of finite rank $⩽n$

We study topological properties of the symmetric inverse topological semigroup of finite transformations ${\mathcal{I}}_{\lambda }^n$ of the rank $⩽n$. We show that the topological inverse semigroup ${\mathcal{I}}_{\lambda }^n$ is algebraically $h$-closed in the class of topological inverse semigroups. Also we prove that a topological semigroup $S$ with countably compact square $S\times S$ does not contain the semigroup ${\mathcal{I}}_{\lambda }^n$ for infinite cardinal $\lambda$ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations ${\mathcal{I}}_{\lambda }^n$ of the rank $⩽n$ is the trivial semigroup.