Given two semigroups $\langle A\rangle$ and $\langle B\rangle$ in ${\mathbb N}^n$, we wonder when they can be glued, i.e., when there exists a semigroup $\langle C\rangle$ in ${\mathbb N}^n$ such that the defining ideals of the corresponding semigroup rings satisfy that $I_C=I_A+I_B+\langle\rho\rangle$ for some binomial $\rho$. If $n\geq 2$ and $k[A]$ and $k[B]$ are Cohen-Macaulay, we prove that in order to glue them, one of the two semigroups must be degenerate. Then we study the two most degenerate cases: when one of the semigroups is generated by one single element (simple split) and the case where it is generated by at least two elements and all the elements of the semigroup lie on a line. In both cases we characterize the semigroups that can be glued and say how to glue them. Further, in these cases, we conclude that the glued $\langle C\rangle$ is Cohen-Macaulay if and only if both $\langle A\rangle$ and $\langle B\rangle$ are also Cohen-Macaulay. As an application, we characterize precisely the Cohen-Macaulay semigroups that can be glued when $n=2$.

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粘合半群：何时以及如何

给定 ${\mathbb N}^n$ 中的两个半群 $\langle A\rangle$ 和 $\langle B\rangle$，我们想知道它们何时可以粘合，即，何时存在半群 $\langle C\rangle $ in ${\mathbb N}^n$ 使得对应半群环的定义理想满足 $I_C=I_A+I_B+\langle\rho\rangle$ 对于某些二项式 $\rho$。如果 $n\geq 2$ 和 $k[A]$ 和 $k[B]$ 是 Cohen-Macaulay，我们证明为了粘合它们，两个半群之一必须是退化的。然后我们研究两种最退化的情况：当一个半群由一个单一元素生成（简单分裂）和由至少两个元素生成并且半群的所有元素位于一条线上的情况。在这两种情况下，我们都刻画了可以粘合的半群并说明如何粘合它们。此外，在这些情况下，我们得出结论，粘合的 $\langle C\rangle$ 是 Cohen-Macaulay 当且仅当 $\langle A\rangle$ 和 $\langle B\rangle$ 也是 Cohen-Macaulay。作为一个应用，我们精确地刻画了当 $n=2$ 时可以粘合的 Cohen-Macaulay 半群。