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On monoids of monotone injective partial selfmaps of integers with cofinite domains and images
Gutik, Oleg ; Repovš, Dušan, 1954-
We study the semigroup ▫$\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$▫ of monotone injective partial selfmaps of the set of integers having cofinite domain and image. We show that ...▫$\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$▫ is bisimple and all of its non-trivial semigroup homomorphisms are either isomorphisms or group homomorphisms. We also prove that every Baire topology ▫$\tau$ on $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$▫ such that ▫$(\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z}),\tau)$▫ is a Hausdorff semitopological semigroup is discrete and we construct a non-discrete Hausdorff inverse semigroup topology ▫$\tau_W$▫ on ▫$\mathscr{I}^\nearrow_\infty (\mathbb{Z})$▫. We show that the discrete semigroup ▫$\mathscr{I}^{\nearrow}_{\infty} (\mathbb{Z})$▫ cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup ▫$S$▫ is an ideal in ▫$S$▫.
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