Shifts of generators and delta sets of numerical monoids.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1315.20054
DOI: 10.1142/S0218196714500271
author: Nathan Kaplan
author: Tyler Lemburg
author: Andrew Niles
author: Christina Zlogar
author: Scott T. Chapman
published in: International Journal of Algebra and Computation
publication date: 2014-10-15T00:00:00Z
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 6356168
zbMATH Keywords: numerical semigroups
zbMATH Keywords: numerical monoids
zbMATH Keywords: Delta sets of factorizations
zbMATH Keywords: non-unique factorizations
zbMATH Keywords: factorization lengths
MaRDI profile type: Publication
cites work: ON DELTA SETS OF NUMERICAL MONOIDS
cites work: Q4951130
cites work: On the arithmetic of certain not integrally closed noetheian integral domains
full work available at URL: s0218196714500271
OpenAlex ID: W1999770809
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title: Shifts of generators and delta sets of numerical monoids.
review text: Let \(a_1,\ldots,a_t\) be a set of \(t\) positive integers. A factorization of \(s\) in terms of \(a_1,\ldots,a_t\) is an expression of the form \(s=z_1a_1+\cdots+z_ta_t\), with \(z_1,\ldots,z_t\) nonnegative integers. The length of this factorization is \(\sum_{i=1}^tz_i\). The set of all lengths of factorizations of \(s\) with respect to \(a_1,\ldots,a_t\) is finite, and so we can write it as \(\{l_1<\cdots<l_k\}\). The Delta set of factorizations of \(s\) is defined as \(\Delta(s)=\{l_2-l_1,l_3-l_2,\ldots,l_k-l_{k-1}\}\).NEWLINENEWLINE For \(S\) the monoid generated by \(\{a_1,\ldots,a_t\}\) (that is, the set of integers \(s\) having at least one factorization in terms of \(a_1,\ldots,a_t\)), we define \(\Delta(S)=\bigcup_{s\in S}\Delta(s)\). For \(t>1\), \(\Delta(S)\) is not empty, and so the simplest Delta set we can obtain is a singleton.NEWLINENEWLINE Write the set \(\{a_1,\ldots,a_t\}\) as \(\{n,n+r_1,\ldots,n+r_{t-1}\}\), with \(r_1<\cdots<r_{t-1}\). The authors prove that fixed \(r_1,\ldots,r_{t-1}\), for \(n\) larger than a constant \(N\) (depending on \(r_{t-1}\) and \(t\)), the set \(\Delta(S)\) is a singleton. This bound is improved for embedding dimension three, that is \(t=2\) and no \(a_i\) has a factorization in terms of the rest. These results have another reading. For \(a_1<\cdots<a_t\), if \(a_t\) is close enough to \(a_1\), then we obtain a monoid with the simplest possible Delta set.
reviewed by: Pedro A. García Sánchez
description: scientific article; zbMATH DE number 6356168