Numerical Semigroups via Projections and via Quotients

Abstract

We examine two natural operations to create numerical semigroups. We say that a numerical semigroup is k-normalescent if it is the projection of the set of integer points in a k-dimensional polyhedral cone, and we say that 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 -quotient and a -quotient is a -quotient. In addition, we prove several results about when a numerical semigroup is not k-normalescent.

Publisher

Springer

Publication Date

4-15-2024

Publication Title

Discrete & Computational Geometry

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1007/s00454-024-00643-z

Language

English

Format

text

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