On the asymptotic behavior of unions of sets of lengths in atomic monoids.
type: Item
instance of: scholarly article
title: On the asymptotic behavior of unions of sets of lengths in atomic monoids.
zbMATH Open document ID: 1154.20049
author: Paul Baginski
author: Scott T. Chapman
author: Natalie Hine
author: João Paixão
published in: Involve
publication date: 2008-09-04T00:00:00Z
review text: Let \(M\) be a commutative multiplicative cancellative atomic monoid. For \(a\in M^*\) let \(L(a)\subset\mathbb{N}\) denote the set of all lengths of factorizations of \(a\) into atoms. For \(n\in\mathbb{N}\), let \(\mathcal V(n)\) be the union of all sets \(L(a)\) for \(a\in M^*\) with \(n\in L(a)\) and \(\Phi(n)=|\mathcal V(n)|\). Let \(\rho(M)\) denote the elasticity of \(M\) (for basic concepts and results of factorization theory see [\textit{A. Geroldinger} and \textit{F. Halter-Koch}, Non-unique factorizations. Algebraic, combinatorial and analytic theory. Pure Appl. Math. 278. Boca Raton: Chapman \& Hall/CRC (2006; Zbl 1113.11002)]). If \(\rho(M)<\infty\), then \(\Phi(n)<\infty\) for all \(n\in\mathbb{N}\), and the authors provide upper and lower bounds for the \(\liminf\) and the \(\limsup\) of \(\Phi(n)/n\) in terms of the maximal and minimal size of gaps in some set \(\mathcal V(n)\). If \(M\) is a Krull monoid with finite class group \(G\) such that every class contains a prime, there is a more precise result concerning the limit of \(\Phi(n)/n\) in terms of the Davenport constant of \(G\) [\textit{S. T. Chapman} and \textit{W. W. Smith}, J. Number Theory 43, No. 1, 24-30 (1993; Zbl 0765.11043)].
reviewed by: Franz Halter-Koch
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 5320744
zbMATH Keywords: nonunique factorizations
zbMATH Keywords: elasticities of factorizations
zbMATH Keywords: unions of sets of lengths
zbMATH Keywords: commutative cancellative atomic monoids
zbMATH Keywords: irreducible elements
zbMATH Keywords: lengths of factorizations
MaRDI profile type: Publication
full work available at URL: involve.2008.1.101
OpenAlex ID: W2132362283
DOI: 10.2140/INVOLVE.2008.1.101
description: scientific article; zbMATH DE number 5320744