# The semigroup of star partial homeomorphisms of a finite deminsional Euclidean space

## TL;DR

This paper introduces and analyzes the algebraic structure of the semigroup of star partial homeomorphisms on finite-dimensional Euclidean spaces, revealing it as a bisimple inverse semigroup with specific congruence properties.

## Contribution

It defines the semigroup of star partial homeomorphisms on al R^n and describes its structure, band, Green's relations, and congruences, which was not previously known.

## Key findings

- The semigroup al PStH_{al R^n} is bisimple and inverse.
- The band of the semigroup is characterized.
- Every non-unit congruence on the semigroup is a group congruence.

## Abstract

In the paper the notion of a star partial homeomorphism of a finite dimensional Euclidean space $\mathbb{R}^n$ is introduced. We describe the structure of the semigroup $\mathbf{PStH}_{\mathbb{R}^n}$ of star partial homeomorphisms of the space $\mathbb{R}^n.$ The structure of the band of $\mathbf{PStH}_{\mathbb{R}^n}$ and Green's relations on $\mathbf{PStH}_{\mathbb{R}^n}$ are described. We show that $\mathbf{PStH}_{\mathbb{R}^n}$ is a bisimple inverse semigroup and every non-unit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.