  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.
