论文标题
在$ \ mathbb {n}^n $中粘合半群和后果
On gluing semigroups in $\mathbb{N}^n$ and the consequences
论文作者
论文摘要
$ \ mathbb {n}^n $ in $ \ mathbb {n}^n $的semigroup $ \ langle c \ rangle $是$ \ langle a \ rangle $和$ \ langle b \ rangle $的粘合定义相应的半群环的理想满足$ i_c $由$ i_a+i_b $和一个额外元素生成的。 Two semigroups $\langle A\rangle$ and $\langle B\rangle$ can be glued if there exist positive integers $k_1,k_2$ such that, for $C=k_1A\sqcup k_2B$, $\langle C\rangle$ is a gluing of $\langle A\rangle$ and $\langle b \ rangle $。尽管任何两个数字半群，即尺寸$ n = 1 $的半群，总是可以粘合，但在较高维度中不再是这种情况。在本文中，我们为$ \ langle a \ rangle $和$ \ langle b \ rangle $的粘合提供了必要的条件，以$ a $ a和$ b $为$ a $ and $ b $，并举例说明为什么它们是必要的。这些概括并解释了先前有关粘合的已知结果。我们还证明，胶合的semigroup $ \ langle c \ rangle $从两个部分$ \ langle a \ rangle $和$ \ langle b \ rangle $中继承了Gorenstein或Cohen-Macaulay等属性。
A semigroup $\langle C\rangle$ in $\mathbb{N}^n$ is a gluing of $\langle A\rangle$ and $\langle B\rangle$ if its finite set of generators $C$ splits into two parts, $C=k_1A\sqcup k_2B$ with $k_1,k_2\geq 1$, and the defining ideals of the corresponding semigroup rings satisfy that $I_C$ is generated by $I_A+I_B$ and one extra element. Two semigroups $\langle A\rangle$ and $\langle B\rangle$ can be glued if there exist positive integers $k_1,k_2$ such that, for $C=k_1A\sqcup k_2B$, $\langle C\rangle$ is a gluing of $\langle A\rangle$ and $\langle B\rangle$. Although any two numerical semigroups, namely semigroups in dimension $n=1$, can always be glued, it is no longer the case in higher dimensions. In this paper, we give necessary and sufficient conditions on $A$ and $B$ for the existence of a gluing of $\langle A\rangle$ and $\langle B\rangle$, and give examples to illustrate why they are necessary. These generalize and explain the previous known results on existence of gluing. We also prove that the glued semigroup $\langle C\rangle$ inherits the properties like Gorenstein or Cohen-Macaulay from the two parts $\langle A\rangle$ and $\langle B\rangle$.
