On monoids of monotone injective partial self-maps of integers with cofinite domains and images

Abstract

We study the semigroup ${\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{Z})$ of monotone injective partial selfmaps of the set of integers having cofinite domain and image. We show that ${\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{Z})$ 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 ${\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{Z})$ such that $({\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{Z}),\tau )$ is a Hausdorff semitopological semigroup is discrete and we construct a non-discrete Hausdorff inverse semigroup topology ${\tau }_W$ on ${\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{Z})$. We show that the discrete semigroup ${\mathcal{I}}_{\infty }^{\nearrow }(\mathbb{Z})$ 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 $S$.

arXiv admin note: text overlap with arXiv:1006.4873

Cited by in corpus (2)

• On the semigroup ${\text{ID}}_{\infty }$
• The monoid of monotone injective partial selfmaps of the poset $({\mathbb{N}}^3,⩽)$ with cofinite domains and images