One half of a maximal embedding dimension numerical semigroup

Abstract

Let S be a numerical semigroup and let p be a positive integer. Then S p = {x ∈ N | px ∈ S} is also a numerical semigroup and when p = 2 we say that S 2 is the one half of the numerical semigroup S. We characterize the numerical semigroups that are one half of a numerical semigroup with maximal embedding dimension. This characterization allows us to algorithmically determine, whether or not a given numerical semigroup is one half of a numerical semigroup with maximal embedding dimension.