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 by Bras-Amor\'os and Bulygin (Semigroup Forum, 79:561--574, 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. Bras-Amor\'os and Bulygin (Semigroup Forum, 79:561--574, 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 multiplicity 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。本文研究了由Bras-Amorós和Bulygin首次引入（Semigroup Forum, 79:561--574, 2009）的半群树中的数值半群无限链。计算结果表明这些链较为罕见，但若没有它们，该树将不会是无限的。我们证明了对于每个亏格$g\geq 5$，不属于无限链的该亏格半群数量多于属于无限链的半群数量。Bras-Amorós和Bulygin（Semigroup Forum, 79:561--574, 2009）给出了属于无限链的半群的一个刻画，该刻画基于半群左元素的互质性，并给出了一个关于数值半群所属无限链集合的基数结果，该结果基于这些左元素的最大公约数的素性。我们重新审视了这些结果，并修正了当左元素的最大公约数为素数时，关于半群所属无限链集合基数的一个不精确之处。随后，我们考察了具有固定重数的子树中的无限链。当重数为素数时，在具有该重数的半群树中仅存在一条无限链。当重数为$4$或$6$时，我们证明了子树中的一种自复制行为，并分别推导出了具有给定亏格和重数$4$及$6$的无限链中半群数量的公式。