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 defining ingredient::derived permutation of $$\sigma_n$$.