# On the monoid of monotone injective partial selfmaps of
$\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images

Research paper by **Oleg Gutik, Inna Pozdniakova**

Indexed on: **26 Jul '16**Published on: **26 Jul '16**Published in: **Mathematics - Group Theory**

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#### Abstract

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
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$.