On the monoid of cofinite partial isometries of ${\mathbb{N}}^n$ with the usual metric

In this paper we study the structure of the monoid $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n$ of cofinite partial isometries of the $n$-th power of the set of positive integers $\mathbb{N}$ with the usual metric for a positive integer $n⩾2$. We describe the elements of the monoid $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n$ as partial transformation of ${\mathbb{N}}^n$, the group of units and the subset of idempotents of the semigroup $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n$, the natural partial order and Green’s relations on $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n$. In particular we show that the quotient semigroup $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n\mathrm{/}{\mathfrak{C}}_{\mathsf{\text{mg}}}$, where ${\mathfrak{C}}_{\mathsf{\text{mg}}}$ is the minimum group congruence on $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n$, is isomorphic to the symmetric group ${\mathcal{S}}_n$ and $\mathcal{D}=\mathcal{J}$ in $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n$. Also, we prove that for any integer $n⩾2$ the semigroup $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}^n$ is isomorphic to the semidirect product ${\mathcal{S}}_n{⋉}_{\mathfrak{h}}({\mathcal{P}}_{\mathrm{\infty }}({\mathbb{N}}^n),\cup )$ of the free semilattice with the unit $({\mathcal{P}}_{\mathrm{\infty }}({\mathbb{N}}^n),\cup )$ by the symmetric group ${\mathcal{S}}_n$.