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

TL;DR

This paper investigates the topological properties of symmetric inverse semigroups of finite rank, demonstrating their algebraic $h$-closedness, restrictions on embedding in certain semigroups, and triviality of their Bohr compactification.

Contribution

It establishes the algebraic $h$-closedness of $ ext{I}_ ext{lambda}^n$, shows non-embedding in countably compact semigroups, and characterizes their Bohr compactification as trivial.

Findings

$ ext{I}_ ext{lambda}^n$ is algebraically $h$-closed in topological inverse semigroups.

No countably compact square semigroup contains $ ext{I}_ ext{lambda}^n$ for infinite $ ext{lambda}$.

The Bohr compactification of $ ext{I}_ ext{lambda}^n$ is trivial.

Abstract

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