# Extensions of semigroups by symmetric inverse semigroups of a bounded finite rank

## TL;DR

This paper investigates the algebraic and topological properties of semigroup extensions formed by symmetric inverse semigroups of bounded finite rank, describing their structure, regularity, and topological extensions.

## Contribution

It provides a detailed description of the structure, regularity, and Green's relations of these semigroup extensions, and introduces the concept of strongly tight ideal series.

## Key findings

- Semigroup extensions are regular, orthodox, inverse, or stable iff the base semigroup has these properties.
- Green's relations are characterized for these extensions.
- Existence of unique compact topological extensions for compact Hausdorff monoids.

## Abstract

We study the semigroup extension $\mathscr{I}_\lambda^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_\lambda^n(S)$ and $\overline{\mathscr{I}_\lambda^n}(S)$ show that the semigroup $\mathscr{I}_\lambda^n(S)$ ($\overline{\mathscr{I}_\lambda^n}(S)$) is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup $\mathscr{I}_\lambda^n(S)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and proved that for any infinite cardinal $\lambda$ and any positive integer $n$ the semigroup $\mathscr{I}_\lambda^n(S)$ has a strongly tight ideal series provides so has $S$. At the finish we show that for every compact Hausdorff semitopological monoid $(S,\tau_S)$ there exists a unique its compact topological extension $\left(\mathscr{I}_\lambda^n(S),\tau_{\mathscr{I}}^\mathbf{c}\right)$ in the class of Haudorff semitopological semigroups.