# On the semigroup $\textbf{ID}_{\infty}$

Research paper by **Oleg Gutik, Anatolii Savchuk**

Indexed on: **16 Apr '19**Published on: **14 Apr '19**Published in: **arXiv - Mathematics - Group Theory**

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#### Abstract

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 $\tau$ on $\textbf{{ID}}_{\infty}$ when $\tau$ 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.