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

In this paper we study the semigroup$mathscr{I}_{infty}^{,Rsh!!!earrow}(mathbb{N})$ of partialco-finite almost monotone bijective transformations of the set ofpositive integers $mathbb{N}$. We show that the semigroup$mathscr{I}_{infty}^{,Rsh!!!earrow}(mathbb{N})$ hasalgebraic properties similar to the bicyclic semigroup: it isbisimple and all of its non-trivial group homomorphisms are eitherisomorphisms or group homomorphisms. Also we prove that every Bairetopology $au$ on$mathscr{I}_{infty}^{,Rsh!!!earrow}(mathbb{N})$ such that$(mathscr{I}_{infty}^{,Rsh!!!earrow}(mathbb{N}),au)$ isa semitopological semigroup is discrete, describe the closure of$(mathscr{I}_{infty}^{,Rsh!!!earrow}(mathbb{N}),au)$ ina topological semigroup and construct non-discrete Hausdorffsemigroup topologies on$mathscr{I}_{infty}^{,Rsh!!!earrow}(mathbb{N})$