Virtual permutation

Definition

Consider the set $\mathbb{N}$ of natural numbers. A virtual permutation of $\mathbb{N}$ is a sequence ${\sigma }_n,n\in \mathbb{N}$ where each ${\sigma }_n$ is a permutation on $\{1,2,\dots ,n\}$ and ${\sigma }_{n-1}$ is the derived permutation of ${\sigma }_n$.