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

TL;DR

This paper investigates the algebraic structure of a semigroup of monotone injective partial selfmaps of the ordered set \\mathbb{N}^2 with cofinite domains and images, revealing its properties, subgroup structure, and Green relations.

Contribution

It characterizes the semigroup of such partial maps, describes its units, idempotents, and Green relations, and establishes the isomorphism of its group of units to a cyclic group of order two.

Findings

The group of units is isomorphic to the cyclic group of order two.

The subsemigroup of idempotents is explicitly described.

Green relations satisfy \\mathscr{D} = \\mathscr{J} in this semigroup.

Abstract

Let $\mathbb{N}^{2}_{\leqslant}$ be the set $\mathbb{N}^{2}$ with the partial order defined as the 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 properties of elements of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ as monotone partial bijections of $\mathbb{N}^{2}_{\leqslant}$ and show that the group of units of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ is isomorphic to the cyclic group of order two. Also we describe the subsemigroup of idempotents of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ and the Green relations on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. In particular, we show that $\mathscr{D}=\mathscr{J}$ in $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$.