## 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 $$c_2n_2\mid c_1n_1$$.

### MSC:

 20M13 Arithmetic theory of semigroups 20M14 Commutative semigroups 13A05 Divisibility and factorizations in commutative rings 20M05 Free semigroups, generators and relations, word problems

Zbl 1279.20072
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