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

TL;DR

This paper investigates the algebraic and topological properties of a semigroup of monotone injective partial maps of integers with cofinite domain and image, revealing its bisimplicity, homomorphism structure, and topological constraints.

Contribution

It establishes the bisimplicity of the semigroup, characterizes its homomorphisms, and analyzes the topological structures, including the non-existence of non-discrete Hausdorff semigroup topologies.

Findings

The semigroup is bisimple.

All non-trivial homomorphisms are either isomorphisms or group homomorphisms.

Discrete topology is the only Hausdorff topology compatible with semitopological structure.

Abstract

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 $S$.