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

TL;DR

This paper investigates the algebraic structure of the monoid of all partial cofinite isometries of positive integers, revealing non-embeddability into a similar monoid on integers, and characterizing its homomorphisms and generating sets.

Contribution

It establishes that the monoid of partial cofinite isometries of positive integers cannot embed into that of integers, and describes the structure of its non-trivial homomorphisms and generators.

Findings

The monoid does not embed into the monoid of all partial cofinite isometries of integers.

Every non-annihilating homomorphism has an image isomorphic to either Z_2 or Z.

The monoid is not finitely generated and lacks a minimal generating set.

Abstract

In the paper we show that the monoid $\mathbf{I}\mathbb{N}_{\infty}$ of all partial cofinite isometries of positive integers does not embed isomorphically into the monoid $\mathbf{ID}_{\infty}$ of all partial cofinite isometries of integers. Moreover, every non-annihilating homomorphism $\mathfrak{h}\colon \mathbf{I}\mathbb{N}_{\infty}\to\mathbf{ID}_{\infty}$ has the following property: the image $(\mathbf{I}\mathbb{N}_{\infty})\mathfrak{h}$ is isomorphic either to the two-element cyclic group $\mathbb{Z}_2$ or to the additive group of integers $\mathbb{Z}(+)$. Also we prove that the monoid $\mathbf{I}\mathbb{N}_{\infty}$ is not finitely generated, and, moreover, monoid $\mathbf{I}\mathbb{N}_{\infty}$ does not contain a minimal generating set.