On the semigroup ID∞\textbf{ID}_{\infty}ID∞​ (original) (raw)

We study the semigroup textbfIDinfty\textbf{{ID}}_{\infty}textbfIDinfty of all partial isometries of the set of integers mathbbZ\mathbb{Z}mathbbZ. It is proved that the quotient semigroup textbfIDinfty/mathfrakCtextsfmg\textbf{{ID}}_{\infty}/\mathfrak{C}_{\textsf{mg}}textbfIDinfty/mathfrakCtextsfmg, where mathfrakCtextsfmg\mathfrak{C}_{\textsf{mg}}mathfrakCtextsfmg is the minimum group congruence, is isomorphic to the group textsfIso(mathbbZ){\textsf{Iso}}(\mathbb{Z})textsfIso(mathbbZ) of all isometries of mathbbZ\mathbb{Z}mathbbZ, textbfIDinfty\textbf{{ID}}_{\infty}textbfIDinfty is an FFF-inverse semigroup, and textbfIDinfty\textbf{{ID}}_{\infty}textbfIDinfty is isomorphic to the semidirect product textsfIso(mathbbZ)ltimesmathfrakhmathscrP!infty(mathbbZ){\textsf{Iso}}(\mathbb{Z})\ltimes_\mathfrak{h}\mathscr{P}_{\!\infty}(\mathbb{Z})textsfIso(mathbbZ)ltimesmathfrakhmathscrP!infty(mathbbZ) of the free semilattice with unit (mathscrP!infty(mathbbZ),cup)(\mathscr{P}_{\!\infty}(\mathbb{Z}),\cup)(mathscrP!infty(mathbbZ),cup) by the group textsfIso(mathbbZ){\textsf{Iso}}(\mathbb{Z})textsfIso(mathbbZ). We give the sufficient conditions on a shift-continuous topology tau\tautau on textbfIDinfty\textbf{{ID}}_{\infty}textbfIDinfty when tau\tautau is discrete. A non-discrete Hausdorff semigroup topology on textbfIDinfty\textbf{{ID}}_{\infty}textbfIDinfty is constructed. Also, the problem of an embedding of the discrete semigroup textbfIDinfty\textbf{{ID}}_{\infty}textbfIDinfty into Hau...