有限阿贝尔群中的完美基
On Perfect Bases in Finite Abelian Groups
摘要：
设$G$是有限阿贝尔群，$s$是正整数. $G$的子集$A$称为{ \em完美$s$ -$G$的基}，如果$G$的每个元素可以唯一地写为至多$s$的和（不一定是不同的）$A$元素;类似地，我们说$A$是{ \em完美限制$s$ -$G$的基}如果$G$的每个元素可以唯一地写为$A$的至多$s$个不同元素的和。我们证明了完美$s$ -基只存在于$s=1$或$|A|=1$的平凡情形中.情况是不同的限制除了完善是更频繁的;在这里，我们对待的情况下，$s=2$和证明$G$有一个完美的限制$2$ -基础，当且仅当，它是同构的$\mathbb{Z}_2$，$\mathbb{Z}_4$，$\mathbb{Z}_7$，$\mathbb{Z}_2^2$，$\mathbb{Z}_2^4$，或$\mathbb{Z}_2^2 \times \mathbb{Z}_4$。
英文摘要：
Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of $A$; similarly, we say that $A$ is a {\em perfect restricted $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ distinct elements of $A$. We prove that perfect $s$-bases exist only in the trivial cases of $s=1$ or $|A|=1$. The situation is different with restricted addition where perfection is more frequent; here we treat the case of $s=2$ and prove that $G$ has a perfect restricted $2$-basis if, and only if, it is isomorphic to $\mathbb{Z}_2$, $\mathbb{Z}_4$, $\mathbb{Z}_7$, $\mathbb{Z}_2^2$, $\mathbb{Z}_2^4$, or $\mathbb{Z}_2^2 \times \mathbb{Z}_4$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
