Erdős-Zaks all divisor sets.
type: Item
instance of: scholarly article
title: Erdős-Zaks all divisor sets.
zbMATH Open document ID: 1263.20056
author: William W. Smith
author: Scott T. Chapman
published in: Periodica Mathematica Hungarica
publication date: 2012-09-17T00:00:00Z
review text: Let \(G\) be an Abelian group, and \(S\) be a subset of \(G\). Let \(\mathfrak F(G,S)\) be the free Abelian group with basis \(S\), and \(\sigma\colon\mathfrak F(G,S)\to G\) be the monoid homomorphism mapping each element of \(S\) to itself. The block monoid \(\mathfrak B(G,S)\) is defined as \(\ker\sigma\). The authors study factorization properties of the monoid \(\mathfrak B(G,S)\), where \(G\) is a finite cyclic group. An all divisor set in \(\mathbb Z_n\) is a subset of \(\mathbb Z_n\) containing \(1\) such that the least residues of the elements in \(S\) are divisors of \(n\) in \(\mathbb Z\). Using Erdős-Zaks' work, the authors develop an algorithm that starts with an `EZADS' input (see title), and leads to a list of blocks which include all the non-primary irreducible blocks in \(\mathfrak B(\mathbb Z_q,S)\): here \(q\) is a positive integer, and \(S\) is an all divisor set (called an EZADS), both of them obtained from the EZADS input. The authors characterize the atoms in the list of blocks. They prove that the block monoid \(\mathfrak B(\mathbb Z_q,S)\) is weakly half-factorial, and obtain a criterion for half-factoriality of \(\mathfrak B(\mathbb Z_q,S)\) using the above mentioned list of blocks. Further results and numerical examples are included.
reviewed by: Moshe Roitman
Mathematics Subject Classification ID: 20M13
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 6083229
zbMATH Keywords: Erdős-Zaks all divisor sets
zbMATH Keywords: atoms
zbMATH Keywords: block monoids
zbMATH Keywords: cross numbers
zbMATH Keywords: EZADS
zbMATH Keywords: non-unique factorizations
zbMATH Keywords: splittable sets
zbMATH Keywords: weakly half-factorial monoids
zbMATH Keywords: algorithms
MaRDI profile type: Publication
full work available at URL: s10998-012-5026-6
OpenAlex ID: W2057210245
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DOI: 10.1007/S10998-012-5026-6
description: scientific article; zbMATH DE number 6083229