Topological monoids of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study the semigroup $I_{\mathrm{\infty }}^{\text{dnearrow}}(N)$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $I_{\mathrm{\infty }}^{\text{dnearrow}}(N)$ 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. Also we prove that every Baire topology $\tau$ on $I_{\mathrm{\infty }}^{\text{dnearrow}}(N)$ such that $(I_{\mathrm{\infty }}^{\text{dnearrow}}(N),\tau )$ is a semitopological semigroup is discrete, describe the closure of $(I_{\mathrm{\infty }}^{\text{dnearrow}}(N),\tau )$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $I_{\mathrm{\infty }}^{\text{dnearrow}}(N)$.