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Symmetric inverse topological semigroups of finite rank $\leqslant n$

We study topological properties of the symmetric inverse topological semigroup of finite transformations $\mathscr{I}_λ^n$ of the rank $\leqslant n$. We show that the topological inverse semigroup $\mathscr{I}_λ^n$ 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 $\mathscr{I}_λ^n$ for infinite cardinal $λ$ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations $\mathscr{I}_λ^n$ of the rank $\leqslant n$ is the trivial semigroup.