胶合半群--何时和如何胶合
Gluing semigroups -- when and how
摘要：
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胶合，即，当存在${\mathbb N}^n$中的半群$\langle C\rangle$，使得相应半群环的定义理想对某个二项式$ρ$满足$I_C=I_A+I_B+\langleρ\rangle$时.如果$n\geq 2$和$k[A]$和$k[B]$是Cohen-Macaulay，我们证明了为了使它们粘合，两个半群之一必须是退化的.然后我们研究了两种最退化的情形：一个半群是由一个单元素生成的（简单分裂），另一个半群是由至少两个元素生成的，并且半群的所有元素都在一条直线上。在这两种情况下，我们都刻画了可以胶合的半群，并说明了如何胶合它们。此外，在这些情况下，我们得出结论，胶合的$\langle C\rangle$是Cohen-Macaulay当且仅当$\langle A\rangle$和$\langle B\rangle$也是Cohen-Macaulay。作为应用，我们精确地刻画了当$n=2$时Cohen-Macaulay半群是可胶合的.
英文摘要：
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 $I_C=I_A+I_B+\langleρ\rangle$ for some binomial $ρ$. 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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