On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

TL;DR

This paper characterizes the topological structure of submonoids of partial cofinite isometries of positive integers, showing they are either compact or discrete under certain conditions, extending to semitopological semigroups with compact ideals.

Contribution

It proves that submonoids containing a specific monoid of partial shifts have only compact or discrete Hausdorff locally compact shift-continuous topologies, and extends this to semitopological semigroups with compact ideals.

Findings

Every such submonoid admits only compact or discrete topologies.

The result extends to semitopological semigroups with an adjoined compact ideal.

The monoid generated by partial shifts has a rigid topological structure.

Abstract

Let $\mathscr{C}_\mathbb{N}$ be a monoid which is generated by the partial shift $\alpha\colon n\mapsto n+1$ of the set of positive integers $\mathbb{N}$ and its inverse partial shift $\beta\colon n+1\mapsto n$. In this paper we prove that if $S$ is a submonoid of the monoid $\mathbf{I}\mathbb{N}_{\infty}$ of all partial cofinite isometries of positive integers which contains $\mathscr{C}_\mathbb{N}$ as a submonoid then every Hausdorff locally compact shift-continuous topology on $S$ with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup $S$ with an adjoined compact ideal.