One major problem in the study of numerical semigroups is determining the growth of the semigroup tree. In the present work, infinite chains of numerical semigroups in the semigroup tree, firstly introduced in Bras-Amor\'os and Nulygin (2009), are studied. Computational results show that these chains are rare, but without them the tree would not be infinite. It is proved that for each genus $g\geq 5$ there are more semigroups of that genus not belonging to infinite chains than semigroups belonging. The reference Bras-Amor\'os and Bulygin (2009) presented a characterization of the semigroups that belong to infinite chains in terms of the coprimality of the left elements of the semigroup as well as a result on the cardinality of the set of infinite chains to which a numerical semigroup belongs in terms of the primality of the greatest common divisor of these left elements. We revisit these results and fix an imprecision on the cardinality of the set of infinite chains to which a semigroup belongs in the case when the greatest common divisor of the left elements is a prime number. We then look at infinite chains in subtrees with fixed multiplicity. When the multiplicity is a prime number there is only one infinite chain in the tree of semigroups with such multiplicity. When the multpliplicity is $4$ or $6$ we prove a self-replication behavior in the subtree and prove a formula for the number of semigroups in infinite chains of a given genus and multiplicity $4$ and $6$, respectively.

翻译：数值半群研究中的一个主要问题是确定半群树的增长。本文研究了Bras-Amorós和Nulygin（2009）首次引入的半群树中数值半群的无限链。计算结果表明，这些链虽然罕见，但没有它们，树将不会是无限的。我们证明了对于每个亏格$g\geq 5$，不属于无限链的半群数量多于属于无限链的半群数量。Bras-Amorós和Bulygin（2009）的参考文献给出了无限链中半群的一个刻画，该刻画基于半群左元素互素性质，并得到了一个关于数值半群所属无限链集合基数依赖于这些左元素最大公约数的素性结果。我们重新审视这些结果，并修正了当左元素最大公约数为素数时半群所属无限链集合基数的不精确之处。随后，我们研究了固定多重性的子树中的无限链。当多重性为素数时，具有该多重性的半群树中仅存在一条无限链。当多重性为$4$或$6$时，我们证明了子树中的自复制行为，并分别推导出了给定亏格下多重性为$4$和$6$的无限链中半群数量的公式。