In this paper we study the semigroup $I_\infty^\dnearrow(N)$ of partial
co-finite almost monotone bijective transformations of the set of positive
integers $\mathbb{N}$. We show that the semigroup $I_\infty^\dnearrow(N)$ 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.