Papers › Numerical semigroups via projections and via quotients
Numerical semigroups via projections and via quotients
Tristram Bogart, Christopher O'Neill, Kevin Woods
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We examine two natural operations to create numerical semigroups. We say that a numerical semigroup 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 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 k₁-quotient and a k₂-quotient is a (k₁+k₂)-quotient. In addition, we prove several results about when a numerical semigroup is not k-normalescent.
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