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

Abstract

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

References in corpus (2)

• Embedding the bicyclic semigroup into countably compact topological semigroups
• On countably compact 0-simple topological inverse semigroups

Cited by in corpus (3)

• On locally compact semitopological $0$-bisimple inverse $\omega$-semigroups
• On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero
• The semigroup of partial co-finite isometries of positive integers