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

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 properties of elements of the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ as monotone partial bijections of ${\mathrm{\mathbb{N}}}_{\le }^2$ and show that the group of units of ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ is isomorphic to the cyclic group of order two. Also we describe the subsemigroup of idempotents of ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$ and the Green relations on ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$. In particular, we show that $\mathcal{D}=\mathcal{J}$ in ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^2\right)$.