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

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 ${\mathbf{P}\mathbf{S}\mathbf{t}\mathbf{H}}_{{\mathbb{R}}^n}$ of star partial homeomorphisms of the space ${\mathbb{R}}^n\mathrm{.}$ The structure of the band of ${\mathbf{P}\mathbf{S}\mathbf{t}\mathbf{H}}_{{\mathbb{R}}^n}$ and Green’s relations on ${\mathbf{P}\mathbf{S}\mathbf{t}\mathbf{H}}_{{\mathbb{R}}^n}$ are described. We show that ${\mathbf{P}\mathbf{S}\mathbf{t}\mathbf{H}}_{{\mathbb{R}}^n}$ is a bisimple inverse semigroup and every non-unit congruence on ${\mathbf{P}\mathbf{S}\mathbf{t}\mathbf{H}}_{{\mathbb{R}}^n}$ is a group congruence.