Topological monoids of almost monotone injective co-finite partial selfmaps of the set of positive integers - Chuchman, I.Ya.; Gutik, O.V. (Carpathian Mathematical Publications, 2010)
In this paper we study the semigroup $\mathscr{I}_{∞}^{\,\Rsh\!\!\! earrow}(\mathbb{N})$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{∞}^{\,\Rsh\!\!\! earrow}(\mathbb{N})$ 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 $τ$ on $\mathscr{I}_{∞}^{\,\Rsh\!\!\! earrow}(\mathbb{N})$ such that $(\mathscr{I}_{∞}^{\,\Rsh\!\!\! earrow}(\mathbb{N}),τ)$ is a semitopological semigroup is discrete, describe the closure of $(\mathscr{I}_{∞}^{\,\Rsh\!\!\! earrow}(\mathbb{N}),τ)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $\mathscr{I}_{∞}^{\,\Rsh\!\!\! earrow}(\mathbb{N})$.