Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension
type: Item
instance of: scholarly article
title: Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension
zbMATH Open document ID: 0972.20036
DOI: 10.1090/S0002-9939-01-05819-1
published in: Proceedings of the American Mathematical Society
publication date: 2001-05-14T00:00:00Z
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1597881
zbMATH Keywords: symmetric numerical semigroups
zbMATH Keywords: multiplicities
zbMATH Keywords: embedding dimensions
zbMATH Keywords: Frobenius numbers
zbMATH Keywords: generators
zbMATH Keywords: minimal presentations
author: José Carlos Rosales
MaRDI profile type: Publication
cites work: Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains
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cites work: Q4951130
cites work: AN ALGORITHMIC METHOD TO COMPUTE A MINIMAL RELATION FOR ANY NUMERICAL SEMIGROUP
cites work: On symmetric numerical semigroups
cites work: On numerical semigroups with high embedding dimension
cites work: Numerical semigroups with Apéry sets of unique expression
Wikidata QID: Q58843061
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review text: A numerical semigroup is an additive submonoid of \(\mathbb{N}\) which generates \(\mathbb{Z}\) as a group. For a numerical semigroup \(S\), its Frobenius number, denoted by \(C(S)\), is the greatest integer not in \(S\); and \(S\) is called symmetric if \(z\in\mathbb{Z}-S\) implies \(C(S)-z\in S\). Also, \(S\) has a unique set of minimal generators \(\{n_0<n_1<\cdots<n_p\}\). The integers \(n_0\) and \(p+1\) are called the multiplicity and embedding dimension of \(S\), and are denoted by \(m(S)\) and \(\mu(S)\), respectively. It is known that if \(S\) is a symmetric numerical semigroup and \(m(S)\geq 3\), then \(\mu(S)\leq m(S)-1\). In this paper, for any integers \(m\) and \(e\) such that \(2\leq e\leq m-1\), the author constructs a symmetric numerical semigroup \(S\) with \(m(S)=m\) and \(\mu(S)=e\). He also shows that if \(\mu(S)\geq 3\), then the cardinality of any minimal presentation for such semigroups is \(\mu(S)(\mu(S)-1)/2-1\).
reviewed by: David F. Anderson
description: scientific article; zbMATH DE number 1597881