Modular translations of numerical semigroups.
type: Item
instance of: scholarly article
title: Modular translations of numerical semigroups.
zbMATH Open document ID: 1266.20074
author: Aureliano M. Robles-Pérez
author: José Carlos Rosales
published in: Semigroup Forum
publication date: 2013-03-21T00:00:00Z
review text: Let \(\mathbb N\) denote the set of nonnegative integers. A numerical semigroup is a submonoid of \(\mathbb N\) with finite complement in \(\mathbb N\). Let \(S\) be a numerical semigroup and let \(m\in S\setminus\{0\}\). For \(a\in\mathbb N\), the set \(\mathrm T(S,a,m)=\{s+(as\bmod m)\mid s\in S\}\) is a numerical semigroup if and only if \(a+1\) and \(m\) are coprime. One says that a numerical semigroup \(S'\) is a modular translation of a numerical semigroup \(S\) if \(S'=\mathrm T(S,a,b)\) for some \(m\in S\setminus\{0\}\) and \(a\in\mathbb N\) (with \(\gcd(a+1,m)=1\)). The aim of this manuscript is to describe the main invariants of \(S'\) in terms of those of \(S\), and this is performed with the help of the Apéry sets. The Apéry set of \(m\) in \(S\) is \(\mathrm{Ap}(S,m)=\{s\in S\mid s-m\not\in S\}\). From this set, by using Selmer's formulas, one can compute the maximum integer not belonging to \(S\) (known as its Frobenius number, denoted by \(\mathrm F(S)\)) and the genus of \(S\) (the number of nonnegative integers not in \(S\)). It is well known that \(\mathrm{Ap}(S,m)=\{0=w(0),w(1),\dots,w(m-1)\}\), where \(w(i)\) is the least integer in \(S\) congruent with \(i\) modulo \(m\). The authors show that \(\mathrm{Ap}(\mathrm T(S,a,m),m)=\{w(i)+(ai\bmod m)\mid i\in\{0,\dots,m-1\}\}\). This, together with Selmer's formulas, is used to determine (and give bounds) for the Frobenius number of \(\mathrm T(S,a,m)\) in terms of the Frobenius number of \(S\). In the same way, a formula for the genus of \(\mathrm T(S,a,m)\) is provided, and with this, the authors are able to prove that if \(\gcd(a,m)=d\) and \(a\mathrm F(S)\bmod m=m-d\) (in this case \(\mathrm F(\mathrm T(S,a,m))=\mathrm F(S)+m-d\)), then \(S\) is irreducible if and only if \(\mathrm T(S,a,b)\) is irreducible. Recall that a numerical semigroup is irreducible if it is not the intersection of two numerical semigroups properly containing it. The numerical semigroup \(S\) has a unique minimal system of…
reviewed by: Pedro A. García Sánchez
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 11D07
zbMATH DE Number: 6147887
zbMATH Keywords: numerical semigroups
zbMATH Keywords: pseudo-Frobenius numbers
zbMATH Keywords: embedding dimension
zbMATH Keywords: Frobenius numbers
zbMATH Keywords: Apéry sets
zbMATH Keywords: modular translations
zbMATH Keywords: generating systems
MaRDI profile type: Publication
Wikidata QID: Q58842589
full work available at URL: s00233-012-9372-8
OpenAlex ID: W2060655559
cites work: Q5837528
cites work: Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups
cites work: Irreducible numerical semigroups.
cites work: Numerical semigroups.
DOI: 10.1007/S00233-012-9372-8
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description: scientific article; zbMATH DE number 6147887