Let $\mathbb{ N}^{ 2}_{ \leqslant}$ be the set $\mathbb{ N}^{ 2}$ with the partial order defined as the product of usual order $\leq$ on the set of positive integers $\mathbb{ N}$. We study the semigroup $\mathscr{ P\!O}\!_{ \infty}(\mathbb{ N}^2_{ \leqslant})$ of monotone injective partial selfmaps of $\mathbb{ N}^{ 2}_{ \leqslant}$ having cofinite domain and image. We describe properties of elements of the semigroup $\mathscr{ P\!O}\!_{ \infty}(\mathbb{ N}^2_{ \leqslant})$ as monotone partial bijections of $\mathbb{ N}^{ 2}_{ \leqslant}$ and show that the group of units of $\mathscr{ P\!O}\!_{ \infty}(\mathbb{ N}^2_{ \leqslant})$ is isomorphic to the cyclic group of order two. Also we describe the subsemigroup of idempotents of $\mathscr{ P\!O}\!_{ \infty}(\mathbb{ N}^2_{ \leqslant})$ and the Green relations on $\mathscr{ P\!O}\!_{ \infty}(\mathbb{ N}^2_{ \leqslant})$. In particular, we show that $\mathscr{ D}=\mathscr{ J}$ in $\mathscr{ P\!O}\!_{ \infty}(\mathbb{ N}^2_{ \leqslant})$.
Categories: math.GR