Simple permutations poset

This article studies the poset of simple permutations with respect to the pattern involvement. We specify results on critically indecomposable posets obtained by Schmerl and Trotter to simple permutations and prove that if $\sigma, \pi$ are two simple permutations such that $\pi < \sigma$ then there exists a chain of simple permutations $\sigma^{(0)} = \sigma, \sigma^{(1)},..., \sigma^{(k)}=\pi$ such that $|\sigma^{(i)}| - |\sigma^{(i+1)}| = 1$ - or 2 when permutations are exceptional- and $\sigma^{(i+1)} < \sigma^{(i)}$. This characterization induces an algorithm polynomial in the size of the output to compute the simple permutations in a wreath-closed permutation class.