On the semigroup ${\text{ID}}_{\infty }$

Abstract

We study the semigroup ${\text{ID}}_{\infty }$ of all partial isometries of the set of integers $\mathbb{Z}$. It is proved that the quotient semigroup ${\text{ID}}_{\infty }/{\mathfrak{C}}_{\text{mg}}$, where ${\mathfrak{C}}_{\text{mg}}$ is the minimum group congruence, is isomorphic to the group $\text{Iso}(\mathbb{Z})$ of all isometries of $\mathbb{Z}$, ${\text{ID}}_{\infty }$ is an $F$-inverse semigroup, and ${\text{ID}}_{\infty }$ is isomorphic to the semidirect product $\text{Iso}(\mathbb{Z}){⋉}_{\mathfrak{h}}{\mathcal{P}}_{\infty }(\mathbb{Z})$ of the free semilattice with unit $({\mathcal{P}}_{\infty }(\mathbb{Z}),\cup )$ by the group $\text{Iso}(\mathbb{Z})$. We give the sufficient conditions on a shift-continuous topology $\tau$ on ${\text{ID}}_{\infty }$ when $\tau$ is discrete. A non-discrete Hausdorff semigroup topology on ${\text{ID}}_{\infty }$ is constructed. Also, the problem of an embedding of the discrete semigroup ${\text{ID}}_{\infty }$ into Hausdorff compact-like topological semigroups is studied.

12 pages, in Ukrainian

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