# The semigroup of partial co-finite isometries of positive integers

## TL;DR

This paper investigates the algebraic structure of the semigroup of all partial co-finite isometries of positive integers, describing its relations, congruences, and its connection to the additive group of integers.

## Contribution

It characterizes the semigroup's Green's relations, band, simplicity, and describes its least group congruence and quotient structure, revealing its isomorphism to the integers.

## Key findings

- The semigroup is simple, $E$-unitary, and $F$-inverse.
- The quotient by the least group congruence is isomorphic to the integers.
- A non-group congruence exists, and group congruences are characterized by restrictions on a bicyclic subsemigroup.

## Abstract

The semigroup $\mathbf{I}\mathbb{N}_{\infty}$ of all partial co-finite isometries of positive integers is studied. We describe Green's relations on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$, its band and proved that $\mathbf{I}\mathbb{N}_{\infty}$ is a simple $E$-unitary $F$-inverse semigroup. We described the least group congruence $\mathfrak{C}_{\mathbf{mg}}$ on $\mathbf{I}\mathbb{N}_{\infty}$ and proved that the quotient-semigroup $\mathbf{I}\mathbb{N}_{\infty}/\mathfrak{C}_{\mathbf{mg}}$ is isomorphic to the additive group of integers. An example of a non-group congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is presented. Also we proved that a congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence if and only if its restriction onto an isomorphic copy of the bicyclic semigroup in $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence.