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

In this paper we study the semigroup ${\mathcal{I}}_{\mathrm{\infty }}^{\nearrow }(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup ${\mathcal{I}}_{\mathrm{\infty }}^{\nearrow }(\mathbb{N})$ 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 ${\mathcal{I}}_{\mathrm{\infty }}^{\nearrow }(\mathbb{N})$ such that $({\mathcal{I}}_{\mathrm{\infty }}^{\nearrow }(\mathbb{N}),\tau )$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $({\mathcal{I}}_{\mathrm{\infty }}^{\nearrow }(\mathbb{N}),\tau )$ in a topological semigroup.