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

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

Indexed on: **27 Jan '17**Published on: **27 Jan '17**Published in: **arXiv - Mathematics - Group Theory**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

Let $\mathbb{N}^{2}_{\leqslant}$ be the set $\mathbb{N}^{2}$ with the partial
order defining as a 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 the natural partial order on the semigroup
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. We proved that the
semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ is isomorphic
to the semidirect product
$\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})\rtimes \mathbb{Z}_2$
of the monoid $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})$ of
oriental monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$
with cofinite domains and images by the group $\mathbb{Z}_2$. Also we describe
the congruence $\sigma$ on the semigroup
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$, which is generated by
the natural order $\preccurlyeq$ on the semigroup
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$: $\alpha\sigma\beta$ if
and only if $\alpha$ and $\beta$ are comparable in
$\left(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant}),\preccurlyeq\right)$.
We prove that quotient semigroup
$\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})/\sigma$ is
isomorphic to the free commutative monoid $\mathfrak{AM}_\omega$ over an
infinite countable set and show that quotient semigroup
$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})/\sigma$ is isomorphic to
the semidirect product of the free commutative monoid $\mathfrak{AM}_\omega$
over an infinite countable set by the group $\mathbb{Z}_2$.