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Numerical Semigroups via Projections and via Quotients

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Abstract

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.

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Notes

  1. It is more standard to also require that \(\gcd ({\mathcal {S}})=1\), but we would like to prove our results in greater generality. In Proposition 2.1 and Remark 2.2, we will see that the two options are actually equivalent from our perspective.

  2. The word normalescent is meant to evoke that it is obtained from a normal semigroup via the process of projection; similar variants on the word “normal” tend to have some established meaning.

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Acknowledgements

Tristram Bogart was supported by internal research grant INV-2020-105-2076 from the Faculty of Sciences of the Universidad de los Andes.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia

    Tristram Bogart

  2. Mathematics Department, San Diego State University, San Diego, CA, 92182, USA

    Christopher O’Neill

  3. Department of Mathematics, Oberlin College, Oberlin, OH, 44074, USA

    Kevin Woods

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  1. Tristram Bogart
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  2. Christopher O’Neill
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  3. Kevin Woods
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Correspondence to Christopher O’Neill.

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Bogart, T., O’Neill, C. & Woods, K. Numerical Semigroups via Projections and via Quotients. Discrete Comput Geom (2024). https://doi.org/10.1007/s00454-024-00643-z

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