The monoid of monotone injective partial selfmaps of the poset $({\mathbb{N}}^3,⩽)$ with cofinite domains and images

Let $n$ be a positive integer $⩾2$ and ${\mathbb{N}}_{⩽}^n$ be the $n$-th power of positive integers with the product order of the usual order on $\mathbb{N}$. In the paper we study the semigroup of injective partial monotone selfmaps of ${\mathbb{N}}_{⩽}^n$ with cofinite domains and images. We show that the group of units $H(\mathbb{I})$ of the semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^n)$ is isomorphic to the group ${\mathcal{S}}_n$ of permutations of an $n$-element set, and describe the subsemigroup of idempotents of $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^n)$. Also in the case $n=3$ we describe the property of elements of the semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^3)$ as partial bijections of the poset ${\mathbb{N}}_{⩽}^3$ and Green’s relations on the semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^3)$. In particular we show that $\mathcal{D}=\mathcal{J}$ in $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^3)$.