Classification of numerical 3-semigroups by means of L-shapes. (Q741657)

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 \(W_a=a\), \(W_b=b\).