Numerical semigroups with multiplicity three and four.
type: Item
instance of: scholarly article
title: Numerical semigroups with multiplicity three and four.
zbMATH Open document ID: 1099.20034
author: N. E. Zubov
published in: Semigroup Forum
publication date: 2006-03-22T00:00:00Z
review text: A numerical semigroup \(S\) is an additive submonoid of the set \(\mathbb{N}\) of nonnegative integers such that \(\mathbb{N}-S\) is finite. The multiplicity (resp., Frobenius number) of \(S\) is the least positive integer in \(S\) (resp., largest integer not in \(S\)), and the gender of \(S\) is the cardinality of \(\mathbb{N}-S\). Numerical semigroups with multiplicity two are uniquely determined by their Frobenius number. The author first shows that a numerical semigroup of multiplicity three is uniquely determined by its Frobenius number and gender, and then gives a formula for the number of numerical semigroups with multiplicity three and given Frobenius number or gender. The author next introduces a new invariant, called the ratio of \(S\), as \(n_2\), where \(\{n_1,n_2,\dots,n_k\}\) is the unique minimal set of generators for \(S\). He shows that a numerical semigroup of multiplicity four is uniquely determined by its Frobenius number, gender, and ratio, and then investigates the set of numerical semigroups of multiplicity four with given Frobenius number or gender.
reviewed by: David F. Anderson
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 5014183
zbMATH Keywords: Frobenius numbers
zbMATH Keywords: genders
zbMATH Keywords: numerical semigroups
zbMATH Keywords: multiplicities
zbMATH Keywords: ratios
zbMATH Keywords: minimal generating sets
MaRDI profile type: Publication
full work available at URL: s00233-005-0517-x
OpenAlex ID: W2088396055
Wikidata QID: Q58842867
DOI: 10.1007/S00233-005-0517-X
description: scientific article; zbMATH DE number 5014183