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.
Categories: math.GR