Numerical Semigroups via Projections and via Quotients

We examine two natural operations to create numerical semigroups. We say that a numerical semigroup \({\mathcal {S}}\) is k-normalescent if it is the projection of the set of integer points in a k-dimensional polyhedral cone, and we say that \({\mathcal {S}}\) is a k-quotient if it is the quotient of a numerical semigroup with k generators. We prove that all k-quotients are k-normalescent, and although the converse is false in general, we prove that the projection of the set of integer points in a cone with k extreme rays (possibly lying in a dimension smaller than k) is a k-quotient. The discrete geometric perspective of studying cones is useful for studying k-quotients: in particular, we use it to prove that the sum of a \(k_1\)-quotient and a \(k_2\)-quotient is a \((k_1+k_2)\)-quotient. In addition, we prove several results about when a numerical semigroup is not k-normalescent.