论文标题
在有限的阿贝尔团体中的完美基地上
On Perfect Bases in Finite Abelian Groups
论文作者
论文摘要
让$ g $为有限的阿贝尔集团，$ s $为正整数。如果$ g $的每个元素可以独特地写入$ g $的每个元素，则$ g $的子集$ a $ a $ g $被称为{\ em perfect $ s $ s $ basis of $ g $}。同样，我们说$ a $是{\ em perfect限制的$ s $ basis的$ g $}，如果$ g $的每个元素都可以独特地写成$ s $ a $ a $的最多元素的总和。我们证明，仅在$ s = 1 $或$ | a | = 1 $的微不足道的情况下才存在完美的$ s $基础。情况有所不同，而限制性则更加频繁。在这里，我们将$ s = 2 $的情况对待，并证明$ g $具有完美的$ 2 $ -BASIS，并且只有在$ \ Mathbb {Z} _2 $，$ \ Mathbb {z} _4 $，$ \ Mathbb {Z} _7 $，$ \ nmathbbbb = 2^2 $ \ mathbb {z} _2^4 $，或$ \ mathbb {z} _2^2 \ times \ times \ mathbb {z} _4 $。
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 $\mathbb{Z}_2^2 \times \mathbb{Z}_4$.
