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

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