Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image

TL;DR

This paper investigates the algebraic and topological properties of a semigroup of monotone partial bijections on positive integers, revealing its similarity to the bicyclic semigroup and characterizing its possible topologies.

Contribution

It establishes the algebraic structure of the semigroup, proves the discreteness of certain topologies, and describes the closure in topological semigroups, extending understanding of such algebraic objects.

Findings

The semigroup is bisimple and has properties similar to the bicyclic semigroup.

Any locally compact topology making it a topological inverse semigroup is discrete.

The closure in a topological semigroup can be explicitly described.

Abstract

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup.