A characterization for all interval doubling schemes of the lattice of permutations

The lattice of all permutations on a -element set has been shown to be bounded [5], which is a strong constructive property characterized by the fact that admits what we call an interval doubling scheme. In this paper we characterize all interval doubling schemes of the lattice , a result that gives a nice precision on the bounded nature of the lattice of permutations. This theorem is a direct corollary of two strong properties (Proposition 3 and Theorem 3) that are also given with their proofs.