Numerical semigroups via projections and via quotients

arXiv2306.11564MaRDI QIDQ6510661

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Abstract: We examine two natural operations to create numerical semigroups. We say that a numerical semigroup $mathcalS$ is $k$-normalescent if it is the projection of the set of integer points in a $k$-dimensional cone, and we say that $mathcalS$ 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.

Has companion code repository: https://github.com/kevwoods/normalescense

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