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

In the paper we show that the monoid $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}$ of all partial cofinite isometries of positive integers does not embed isomorphically into the monoid ${\mathbf{I}\mathbf{D}}_{\mathrm{\infty }}$ of all partial cofinite isometries of integers. Moreover, every non-annihilating homomorphism $\mathfrak{h}\colon $\mathbf{I}$$\mathbb{N}${\infty}\to$\mathbf{I}\mathbf{D}${\infty}$ has the following property: the image $(\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }})\mathfrak{h}$ 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 $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}$ is not finitely generated, and, moreover, monoid $\mathbf{I}{\mathbb{N}}_{\mathrm{\infty }}$ does not contain a minimal generating set.