The doubles of a numerical semigroup

Let SS be a numerical semigroup and let pp be a positive integer. Then the quotient Sp={x∈N∣px∈S} is also a numerical semigroup. When p=2p=2 we say that S2 is half of the numerical semigroup SS. Dually, we say that SS is a double of the numerical semigroup S2. We characterize the set of all doubles of a numerical semigroup. We also give some alternative proofs and improvements for some results that we find in previous papers.