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

Abstract

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 $\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)$ of monotone injective partial selfmaps of ${\mathbb{N}}_{⩽}^2$ having cofinite domain and image. We describe properties of elements of the semigroup $\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)$ as monotone partial bijections of ${\mathbb{N}}_{⩽}^2$ and show that the group of units of $\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)$ is isomorphic to the cyclic group of order two. Also we describe the subsemigroup of idempotents of $\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)$ and the Green relations on $\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)$. In particular, we show that $\mathcal{D}=\mathcal{J}$ in $\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)$.

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