On the monoid of cofinite partial isometries of $\mathbb{N}^n$ with the usual metric

TL;DR

This paper investigates the algebraic structure of the monoid of cofinite partial isometries on the n-dimensional positive integer lattice, revealing its decomposition, group relations, and isomorphisms with well-known algebraic objects.

Contribution

It provides a detailed structural analysis of the monoid of cofinite partial isometries on  integer lattices, including descriptions of its elements, substructures, and isomorphisms with semidirect products.

Findings

The quotient of the monoid by the minimum group congruence is isomorphic to the symmetric group S_n.

The semigroup IbN__^n is isomorphic to a semidirect product involving S_n and the semilattice of finite subsets of bN^n.

The semigroup satisfies D=J relations.

Abstract

In this paper we study the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of positive integers $\mathbb{N}$ with the usual metric for a positive integer $n\geqslant 2$. We describe the elements of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ as partial transformation of $\mathbb{N}^n$, the group of units and the subset of idempotents of the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$, the natural partial order and Green's relations on $\mathbf{I}\mathbb{N}_{\infty}^n$. In particular we show that the quotient semigroup $\mathbf{I}\mathbb{N}_{\infty}^n/\mathfrak{C}_{\textsf{mg}}$, where $\mathfrak{C}_{\textsf{mg}}$ is the minimum group congruence on $\mathbf{I}\mathbb{N}_{\infty}^n$, is isomorphic to the symmetric group $\mathscr{S}_n$ and $\mathscr{D}=\mathscr{J}$ in $\mathbf{I}\mathbb{N}_{\infty}^n$. Also, we prove that for any integer $n\geqslant 2$ the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$ is isomorphic to the semidirect product ${\mathscr{S}_n\ltimes_\mathfrak{h}(\mathscr{P}_{\infty}(\mathbb{N}^n),\cup)}$ of the free semilattice with the unit $(\mathscr{P}_{\infty}(\mathbb{N}^n),\cup)$ by the symmetric group $\mathscr{S}_n$.