The monoid of monotone injective partial selfmaps of the poset (ℕ3,≤)with cofinite domains and images

Анотація

Let $n$ be a positive integer $\ge 2$ and ${\mathrm{\mathbb{N}}}_{\le }^n$ be the $n$-th power of positive integers with the product order of the usual order on $\mathrm{\mathbb{N}}$. In the paper we study the semigroup of injective partial monotone selfmaps of ${\mathrm{\mathbb{N}}}_{\le }^n$ with cofinite domains and images. We show that the group of units $H\left(\mathrm{I}\right)$ of the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^n\right)$ is isomorphic to the group ${\mathcal{S}}_n$ of permutations of an $n$-element set, and describe the subsemigroup of idempotents of ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^n\right)$.  Also in the case $n=3$ we describe the property of elements of the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^3\right)$ as partial bijections of the poset ${\mathrm{\mathbb{N}}}_{\le }^3$ and Green's relations on the semigroup ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^3\right)$. In particular we show that $\mathcal{D}=\mathcal{J}$ in ${\mathcal{PO}}_{\infty }\left({\mathrm{\mathbb{N}}}_{\le }^3\right)$.