Classification of numerical 3-semigroups by means of L-shapes.
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 weighted 2-Cayley digraph \(G(c;a,b;a,b)\), i.e. with \(W_a=a\), \(W_b=b\).
scientific article; zbMATH DE number 6343963