A note on feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

TL;DR

This paper investigates the properties of feebly compact topologies on symmetric inverse semigroups of finite transformations, establishing conditions under which these topologies are sequentially pracompact or compact.

Contribution

It characterizes when feebly compact shift-continuous $T_1$-topologies on these semigroups are sequentially pracompact or compact, providing new insights into their topological structure.

Findings

Feebly compact shift-continuous $T_1$-topologies are sequentially pracompact if and only if they are feebly compact.

Every feebly $ ext{omega}$-bounded $T_1$-topology on the semigroup is compact.

The results clarify the relationship between feebly compactness and other compactness properties in this context.

Abstract

We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_\lambda^n$ of finite transformations of the rank $\leqslant n$. It is proved that such $T_1$-topology is sequentially pracompact if and only if it is feebly compact. Also, we show that every shift-continuous feebly $\omega$-bounded $T_1$-topology on $\mathscr{I}_\lambda^n$ is compact.