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**When the catenary degree agrees with the tame degree in numerical semigroups of embedding dimension three.**
*(English)*
Zbl 1326.20061

Let \(S\) be a numerical semigroup (a cofinite submonoid of \((\mathbb N,+)\)), and let \(\{n_1,\ldots,n_p\}\) be the unique minimal generating system of \(S\). A \(p\)-tuple \(x=(x_1,\ldots,x_p)\in\mathbb N^p\) such that \(s=x_1n_1+\cdots+x_pn_p\) is known as a factorization of \(s\). If \(x=(x_1,\ldots,x_p)\) and \(y=(y_1,\ldots,y_p)\) are factorizations of \(s\), the distance between \(x\) and \(y\) is defined as \(d(x,y)=\max\{|x-\gcd(x,y)|,|y-\gcd(x,y)|\}\), where \(\gcd(x,y)=(\min\{x_1,y_1\},\ldots,\min\{x_p,y_p\})\) is the common part of \(x\) and \(y\) and \(|x|=x_1+\cdots+x_p\). Denote the set of factorizations of \(s\) by \(\mathsf Z(s)\).

The catenary degree of \(S\), denoted by \(\mathsf c(S)\), is the least integer \(c\) such that for every \(x,y\) factorizations of an element \(s\in S\), there exists a sequence of factorizations of \(s\) joining them whose consecutive links are closer to \(c\). The tame degree of \(S\), denoted by \(\mathsf t(S)\), is the least integer \(t\) such that for any \(s\in S\) and any factorization \(x\) of \(s\) containing a certain \(n_i\), there exists another factorization \(y\) of \(s\) containing \(n_i\) and such that \(d(x,y)\leq t\). For \(s\in S\) we can build a graph from the factorizations of \(S\) by linking the \(x,y\) such that \(|\gcd(x,y)|\neq 0\); the connected components of this graph are called \(\mathcal R\)-classes. An element of \(S\) is said to be a Betti element if its associated graph has more than one \(\mathcal R\)-class.

In general \(\mathsf c(S)\leq\mathsf t(S)\). In the article under review, the authors investigate the numerical semigroups \(S=\langle n_1,n_2,n_3\rangle\), where \(n_1<n_2<n_3\), for which the equality \(\mathsf c(S)=\mathsf t(S)\) holds. Defining, for \(\{i,j,k\}=\{1,2,3\}\), the integers \(c_i=\min\{K\in\mathbb Z^+\mid Kn_i\in\langle n_j,n_k\rangle\}\), in this setting \(\text{Betti}(S)=\{c_1n_1,c_2n_2,c_3n_3\}\), and thus \(|\text{Betti}(S)|\leq 3\). Since it has been proved by V. Blanco et al. [Ill. J. Math. 55, No. 4, 1385-1414 (2011; Zbl 1279.20072)] that \(\mathsf c(S)=\mathsf t(S)\) if \(|\text{Betti}(S)|=3\), the authors explore whether the equality holds for \(|\text{Betti}(S)|\in\{1,2\}\), by studying the factorizations and \(\mathcal R\)-classes associated to every possible configuration of the Betti elements of \(S\) under this assumption. Their analysis provides a complete characterization for \(3\)-generated numerical semigroups such that \(\mathsf c(S)=\mathsf t(S)\), and shows that the equality holds if and only if \(|\text{Betti}(S)|\neq 2\) or \(c_1n_1\neq c_2n_2=c_3n_3\) and \(c_2n_2\mid c_1n_1\).

The catenary degree of \(S\), denoted by \(\mathsf c(S)\), is the least integer \(c\) such that for every \(x,y\) factorizations of an element \(s\in S\), there exists a sequence of factorizations of \(s\) joining them whose consecutive links are closer to \(c\). The tame degree of \(S\), denoted by \(\mathsf t(S)\), is the least integer \(t\) such that for any \(s\in S\) and any factorization \(x\) of \(s\) containing a certain \(n_i\), there exists another factorization \(y\) of \(s\) containing \(n_i\) and such that \(d(x,y)\leq t\). For \(s\in S\) we can build a graph from the factorizations of \(S\) by linking the \(x,y\) such that \(|\gcd(x,y)|\neq 0\); the connected components of this graph are called \(\mathcal R\)-classes. An element of \(S\) is said to be a Betti element if its associated graph has more than one \(\mathcal R\)-class.

In general \(\mathsf c(S)\leq\mathsf t(S)\). In the article under review, the authors investigate the numerical semigroups \(S=\langle n_1,n_2,n_3\rangle\), where \(n_1<n_2<n_3\), for which the equality \(\mathsf c(S)=\mathsf t(S)\) holds. Defining, for \(\{i,j,k\}=\{1,2,3\}\), the integers \(c_i=\min\{K\in\mathbb Z^+\mid Kn_i\in\langle n_j,n_k\rangle\}\), in this setting \(\text{Betti}(S)=\{c_1n_1,c_2n_2,c_3n_3\}\), and thus \(|\text{Betti}(S)|\leq 3\). Since it has been proved by V. Blanco et al. [Ill. J. Math. 55, No. 4, 1385-1414 (2011; Zbl 1279.20072)] that \(\mathsf c(S)=\mathsf t(S)\) if \(|\text{Betti}(S)|=3\), the authors explore whether the equality holds for \(|\text{Betti}(S)|\in\{1,2\}\), by studying the factorizations and \(\mathcal R\)-classes associated to every possible configuration of the Betti elements of \(S\) under this assumption. Their analysis provides a complete characterization for \(3\)-generated numerical semigroups such that \(\mathsf c(S)=\mathsf t(S)\), and shows that the equality holds if and only if \(|\text{Betti}(S)|\neq 2\) or \(c_1n_1\neq c_2n_2=c_3n_3\) and \(c_2n_2\mid c_1n_1\).

Reviewer: Alessio Moscariello (Pagani)

### MSC:

20M13 | Arithmetic theory of semigroups |

20M14 | Commutative semigroups |

13A05 | Divisibility and factorizations in commutative rings |

20M05 | Free semigroups, generators and relations, word problems |