  Let $n$ be a positive integer $\geqslant 2$ and $\mathbb{N}^n_{\leqslant}$ 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 $\mathbb{N}^n_{\leqslant}$ with cofinite domains and
images. We show that the group of units $H(\mathbb{I})$ of the semigroup
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$ is isomorphic to the
group $\mathscr{S}_n$ of permutations of an $n$-element set, and describe the
subsemigroup of idempotents of
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$. Also in the case $n=3$
we describe the property of elements of the semigroup
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$ as partial bijections of
the poset $\mathbb{N}^3_{\leqslant}$ and Green's relations on the semigroup
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$. In particular we show
that $\mathscr{D}=\mathscr{J}$ in
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$.
