In this paper we study the semigroupmathscrIinfty,Rsh!!!earrowβ(mathbbN) of partialco-finite almost monotone bijective transformations of the set ofpositive integers mathbbN. We show that the semigroupmathscrIinfty,Rsh!!!earrowβ(mathbbN) 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 onmathscrIinfty,Rsh!!!earrowβ(mathbbN) such that(mathscrIinfty,Rsh!!!earrowβ(mathbbN),au) isa semitopological semigroup is discrete, describe the closure of(mathscrIinfty,Rsh!!!earrowβ(mathbbN),au) ina topological semigroup and construct non-discrete Hausdorffsemigroup topologies onmathscrIinfty,Rsh!!!earrowβ(mathbbN)