On the delta set and the Betti elements of a BF-monoid.
type: Item
instance of: scholarly article
title: On the delta set and the Betti elements of a BF-monoid.
zbMATH Open document ID: 1277.20071
author: Yanyan Li
published in: Arabian Journal of Mathematics
publication date: 2012-12-18T00:00:00Z
review text: A monoid \(S\) is called a bounded factorization monoid, or BF-monoid, if for any \(x\in S\) the set of lengths of possible factorizations of \(S\) is bounded. In this paper the authors investigate certain factorization invariants of BF-monoids, focusing on the Delta set of \(S\), denoted \(\Delta(S)\). Let \(x\in S\) and \(\mathcal L(x)=\{m_1<m_2<\cdots<m_k\}\) denote the set of lengths of the factorizations of \(x\). Then \(\Delta(x)=\{m_i-m_{i-1}\mid 2\leq i\leq k\}\). The Delta set of \(S\) is the union of the Delta sets of the elements of \(S\). The Delta set of a BF-monoid \(S\) is one measure of how far \(S\) is from being a unique factorization domain. For example, \(\Delta(S)\) is empty if and only if for each \(x\in S\) all factorizations of \(x\) have the same length, and \(\Delta(S)=\{k\}\) if and only if for each \(x\in S\) the set of lengths of \(x\) is an arithmetic progression of difference \(k\). Factorization properties of BF-monoids can be quite subtle and even monoids that are easy to describe can have complicated Delta sets. The authors show that for a finitely presented BF-monoid \(S\) the minimum and maximum of \(\Delta(S)\) can be determined by focusing on the factorizations of a certain finite subset of the elements of \(S\) known as the Betti elements of \(S\). In the particular case where \(S\) is a numerical monoid with a minimal generating set of size \(3\) they give an explicit description of \(\max\Delta(S)\) in terms of the generators and use it to give a description of when \(\Delta(S)\) consists of a single element.
reviewed by: Nathan Kaplan
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 11D07
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 6116564
zbMATH Keywords: bounded factorization monoids
zbMATH Keywords: numerical monoids
zbMATH Keywords: numerical semigroups
zbMATH Keywords: non-unique factorizations
zbMATH Keywords: lengths of factorizations
zbMATH Keywords: irreducible elements
zbMATH Keywords: irreducible factorizations
zbMATH Keywords: Betti elements
zbMATH Keywords: Delta sets
MaRDI profile type: Publication
full work available at URL: s40065-012-0019-0
OpenAlex ID: W2003298801
Wikidata QID: Q59292247
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DOI: 10.1007/S40065-012-0019-0
description: scientific article; zbMATH DE number 6116564