Numerical semigroups via projections and via quotients
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917638630801408 |
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| author | Bogart, Tristram O'Neill, Christopher Woods, Kevin |
| author_facet | Bogart, Tristram O'Neill, Christopher Woods, Kevin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11564 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Numerical semigroups via projections and via quotients Bogart, Tristram O'Neill, Christopher Woods, Kevin Commutative Algebra Combinatorics 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. |
| title | Numerical semigroups via projections and via quotients |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2306.11564 |