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.

10 pages. in Ukrainian