On the semigroup $\textbf{ID}_{\infty}$
Олег Гутік, Anatolii Savchuk · arXiv (Cornell University) · 2019
We study the semigroup $\textbf{ID}_{\infty}$ of all partial isometries of the set of integers $\mathbb{Z}$. It is proved that the quotient semigroup $\textbf{ID}_{\infty}/\mathfrak{C}_{\textsf{mg}}$, where $\mathfrak{C}_{\textsf{mg}}$ is the minimum group congruence, is isomorphic to the group ${\textsf{Iso}}(\mathbb{Z})$ of all isometries of $\mathbb{Z}$, $\textbf{ID}_{\infty}$ is an $F$-inverse semigroup, and $\textbf{ID}_{\infty}$ is isomorphic to the semidirect product ${\textsf{Iso}}(\mathbb{Z})\ltimes_\mathfrak{h}\mathscr{P}_{\!\infty}(\mathbb{Z})$ of the free semilattice with unit $(\mathscr{P}_{\!\infty}(\mathbb{Z}),\cup)$ by the group ${\textsf{Iso}}(\mathbb{Z})$. We give the sufficient conditions on a shift-continuous topology $τ$ on $\textbf{ID}_{\infty}$ when $τ$ is discrete. A non-discrete Hausdorff semigroup topology on $\textbf{ID}_{\infty}$ is constructed. Also, the problem of an embedding of the discrete semigroup $\textbf{ID}_{\infty}$ into Hausdorff compact-like topological semigroups is studied.