On the monoid of cofinite partial isometries of ℕ with the usual metric

Анотація

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