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

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 $\mathfrak{h}:\mathbf{I}{\mathbb{N}}_{\infty }\to {\mathbf{ID}}_{\infty }$ 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.

11 pages. arXiv admin note: text overlap with arXiv:1909.08823, arXiv:1904.11802