A numerical semigroup, S, is a subsemigroup of N0 that is closed under addition. There is much literature that characterizes several invariants that exist for S, including the Ap ́ery set, factorization sets, and length sets. In this paper we describe a relationship between the numerical semigroup S and two other numerical semigroups SM and Sm that are derived from S. We use these two new numerical semigroups to characterize the maximum and minimum factorization length for an element in S. We then expand on our results and provide a refined version of the structure theorem for sets of length for numerical semigroups.
