On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise

TL;DR

This paper investigates the algebraic and topological properties of a monoid of cofinite partial isometries of positive integers with bounded finite noise, extending classical results to new algebraic structures.

Contribution

It introduces and analyzes the monoid of cofinite partial isometries with bounded noise, providing new results on its topological and algebraic structure, including closure properties.

Findings

Hausdorff shift-continuous topology on the monoid is discrete

The monoid's closure in a topological inverse semigroup has a specific algebraic structure

If embedded densely, the complement forms a closed ideal or a topological group

Abstract

In the paper we study algebraic properties of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ of cofinite partial isometries of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$. For the monoids $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $\tau$ on $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is discrete and if $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a proper dense subsemigroup of a Hausdorff semitopological semigroup $S$, then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a closed ideal of $S$, and moreover if $S$ is a topological inverse semigroup then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a topological group. Also we describe the algebraic and topological structure of the closure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ in a locally compact topological inverse semigroup.