Classification of numerical 3-semigroups by means of L-shapes.
type: Item
instance of: scholarly article
title: Classification of numerical 3-semigroups by means of L-shapes.
zbMATH Open document ID: 1325.20052
author: Francesc Aguiló
author: C. Marijuán
published in: Semigroup Forum
publication date: 2014-09-12T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/28487
review text: Given \(a,b,c\in\mathbb N\), \(1\leq a<b<c\), \(\gcd(a,b,c)=1\), and \(W_a,W_b\in\mathbb R^+\), a \textit{weighted 2-Cayley digraph} \(G(c;a,b;W_a,W_b)\) is the directed graph with vertex set \(\mathbb Z_c\) and arcs \([g]_c\to [g+a]_c\) with weight \(W_a\) for \(g=0,\ldots,c-1\), and \([g]_c\to [g+b]_c\) with weight \(W_b\) for \(g=0,\ldots,c-1\) (where \([\cdot]_c\) denotes the residue class modulo \(c\)). Let \(i,j:=[i,i+1]\times[j,j+1]\subseteq\mathbb R^2\). An \textit{L-shape related to the digraph} \(G(c;a,b;W_a,W_b)\) consists of \(c\) unit squares \(i,j\) such that \((i,j)\in\mathbb N^2\) and the residue classes \([ia+jb]_c\) are all distinct. Let \(\delta(i,j)=iW_a+jW_b\), and given \(n\in\mathbb N\) denote \(Q_n=\{i,j:[ia+jb]_c=[n]_c,\;0\leq i,j\}\). An L-shape \(L\) related to the digraph \(G(c;a,b;W_a,W_b)\) is a \textit{minimum distance diagram} if \(\delta(s,t)=\min\{\delta(i,j):i,j\in Q_{sa+tb}\}\) for each \(s,t\in L\) and if \(u,v\in L\) whenever \(s,t\in L\) and \(0\leq u\leq s\), \(0\leq v\leq t\). In the first part of the paper under review, the authors classify the minimum distance diagrams of a weighted 2-Cayley digraph in terms of the lengths of the sides of a given minimum distance diagram of the digraph. A \textit{numerical semigroup} \(S\) is a cofinite submonoid of \((\mathbb N,+)\). \(S\) is \textit{symmetric} if for every \(n\in\mathbb Z\) we have \(n\in S\) if and only if \(\max(\mathbb Z\setminus S)-n\notin S\). \(S=\langle b_0,\ldots,b_g\rangle\) is called \textit{free} if \(N_ib_i\in\langle b_0,\ldots,b_{i-1}\rangle\) for \(i=1,\ldots,g\) where \(N_i=e_{i-1}/e_i\) with \(e_i=\gcd(b_0,\ldots,b_i)\). \(S\) is a \textit{plane curve semigroup} if \(S\) is free and in addition \(N_ib_i<b_{i+1}\). In the second part of the paper, the authors characterize the above three properties of a three-generated numerical semigroup \(S=\langle a,b,c\rangle\) in terms of the minimum distance diagrams associated to the weigh…
reviewed by: Alessio Sammartano
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 11D07
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 6343963
zbMATH Keywords: weighted 2-Cayley digraphs
zbMATH Keywords: minimum distance diagrams
zbMATH Keywords: three-generated numerical semigroups
zbMATH Keywords: Frobenius numbers
zbMATH Keywords: minimal generating sets
MaRDI profile type: Publication
OpenAlex ID: W1978121070
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DOI: 10.1007/S00233-013-9560-1
description: scientific article; zbMATH DE number 6343963