The semigroup of partial co-finite isometries of positive integers

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.