Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image

Details Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image

2011-08-14

arxiv.org/abs/1108.2848v1

Studia Sci. Math. Hungar. 48:3 (2011), 342-353

Mathematics

General Topology

Scientific paper

10.1556/SScMath.48.2011.3.1176

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ 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. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup.

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