  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 $S$.
