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

Let ${\mathbb{N}}_{⩽}^2$ be the set ${\mathbb{N}}^2$ with the partial order defined as the product of usual order $\le$ on the set of positive integers $\mathbb{N}$. We study the semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^2)$ of monotone injective partial selfmaps of ${\mathbb{N}}_{⩽}^2$ having cofinite domain and image. We describe the natural partial order on $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^2)$ and show that it coincides with the natural partial order which is induced from symmetric inverse monoid ${\mathcal{I}}_{\mathbb{N}\times \mathbb{N}}$ onto $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^2)$. We proved that the semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^2)$ is isomorphic to the semidirect product $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}^{+}({\mathbb{N}}_{⩽}^2)⋊{\mathbb{Z}}_2$ of the monoid $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}^{+}({\mathbb{N}}_{⩽}^2)$ of orientation-preserving monotone injective partial selfmaps of ${\mathbb{N}}_{⩽}^2$ with cofinite domains and images by the cyclic group ${\mathbb{Z}}_2$ of the order two. Also we describe the congruence $\sigma$ on $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^2)$ which is generated by the natural order $\preccurlyeq$ on the semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^2)$. We prove that the quotient semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}^{+}({\mathbb{N}}_{⩽}^2)\mathrm{/}\sigma$ is isomorphic to the free commutative monoid ${\mathfrak{A}\mathfrak{M}}_{\omega }$ over an infinite countable set and show that the quotient semigroup $\mathcal{P}\text{\hspace{0.17em}⁣}\mathcal{O}\text{\hspace{0.17em}⁣}{}_{\mathrm{\infty }}({\mathbb{N}}_{⩽}^2)\mathrm{/}\sigma$ is isomorphic to the semidirect product of the free commutative monoid ${\mathfrak{A}\mathfrak{M}}_{\omega }$ by ${\mathbb{Z}}_2$.