On the monoid of monotone injective partial selfmaps of ℕ≤2 with cofinite domains and images, II

Анотація

Let ${\mathrm{\mathbb{N}}}_{\le }^2$ be the set ${\mathrm{\mathbb{N}}}^2$ with the partial order defined as the product of usual order $\le$ on the set of positive integers $\mathrm{\mathbb{N}}$. We study the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ of monotone injective partial selfmaps of ${\mathrm{\mathbb{N}}}_{\le }^2$ having cofinite domain and image. We describe the natural partial order on the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ and show that it coincides with the natural partial order which is induced from symmetric inverse monoid ${\mathcal{I}}_{\mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}}$ over the set $\mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}$ onto the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$. We proved that the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ is isomorphic to the semidirect product ${\mathcal{PO}}_{\infty }^{+}\left({\mathrm{\mathbb{N}}}_{\le }^2\right)\times {\mathrm{\mathbb{Z}}}_2$ of the monoid ${\mathcal{PO}}_{\infty }^{+}\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ of orientation-preserving monotone injective partial selfmaps of ${\mathrm{\mathbb{N}}}_{\le }^2$ with cofinite domains and images by the cyclic group ${\mathrm{\mathbb{Z}}}_2$ of the order two. Also we describe the congruence $\sigma$ on the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ which is generated by the natural order $⩽$ on the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$: $\alpha \sigma \beta$ if and only if $\alpha$ and $\beta$ are comparable in $\left({\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right),⩽\right)$. We prove that the quotient semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)/\sigma$ is isomorphic to the free commutative monoid ${\mathfrak{AM}}_{\omega }$ over an infinite countable set and show that the quotient semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)/\sigma$ is isomorphic to the semidirect product of the free commutative monoid ${\mathfrak{AM}}_{\omega }$ by the group ${\mathrm{\mathbb{Z}}}_2$.