EZADS inputs which produce half-factorial block monoids.
type: Item
instance of: scholarly article
title: EZADS inputs which produce half-factorial block monoids.
zbMATH Open document ID: 1330.20086
author: Jeffrey A. Manning
published in: Semigroup Forum
publication date: 2015-10-12T00:00:00Z
review text: Let \(A=\{a_0=1,a_1,\ldots,a_n\}\) be a set of pairwise co-prime integers, put \(q=\prod_{i=1}^na_i\), and let \(S=S(A):=\{q_0,q_1,\ldots,q_n\}\subset C_q\), where \(q_i\) is the residue class mod \(q\) containing \(q/a_i\). The block monoid \(B(C_q,S)\) associated with \(S\) is defined as the monoid consisting of all elements \(B=\prod_{s\in S}s^{v_s}\) (with \(v_s\) being non-negative integers), satisfying \(\sum_{s\in S}v_ss=0\). The aim of the author is to describe sets \(A\) for which \(B(C_q,S(A))\) is half-factorial, i.e., for each of its elements \(a\) all factorizations of \(a\) into irreducibles have the same length. He observes (Proposition 2.1) that from a result of \textit{S. T. Chapman} and \textit{W. W. Smith} [Isr. J. Math. 71, No. 1, 65-95 (1990; Zbl 0717.13014)] it follows that for \(n=2\) all monoids \(B(C_q,S)\) are half-factorial, and shows (Theorem 3.2) that if \(A_1\subset A_2\), and \(B(C_q,S(A_2))\) is half-factorial, then also \(B(C_q,S(A_1))\) is half-factorial. This leads to an inductive way of classifying all half-factorial monoids \(B(C_q,S)\).
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 20M13
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 6493124
zbMATH Keywords: block monoids
zbMATH Keywords: half-factorial sets
zbMATH Keywords: cyclic groups
zbMATH Keywords: factorization properties
zbMATH Keywords: cross numbers
MaRDI profile type: Publication
full work available at URL: s00233-015-9698-0
OpenAlex ID: W2017837504
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DOI: 10.1007/S00233-015-9698-0
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description: scientific article; zbMATH DE number 6493124