整数的极大可加性可约子集
Maximally additively reducible subsets of the integers
AI总结：
本文研究整数集合的加法因子，证明了包含0或不包含0的特定区间在对应子集中具有最多因子，并利用 lunar 乘法解决了相关猜想。
摘要：
设 $A, B \subseteq \mathbb{N}$ 为两个有限自然数集合。若存在某个 $C \subseteq \mathbb{N}$ 使得 $A = B+C$，则称 $B$ 是 $A$ 的加法因子。我们证明，在以 $0$ 为元素的 $\{0, 1, \ldots, k\}$ 的子集中，完整区间 $\{0, 1, \ldots,k\}$ 具有最多的因子。为推广至不以 $0$ 为元素的集合，我们证明了加法因子与 lunar multiplication（月球乘法）之间的对应关系，该方法由 Appelgate、LeBrun 和 Sloane (2011) 在研究一种 min/max 算术时引入。二元 lunar 因子的数量与整数拆分相关，此类拆分受限的条件是其第一部分大于或等于所有其他部分。我们建立了此类拆分的若干界，以证明 $\{1, \ldots, k\}$ 在 $\{0, 1, \ldots, k\}$ 的所有子集中具有最多的因子。这些结果解决了 LeBrun 等人关于极大 lunar 二元因子数量的两个猜想，该数量是关于任意进制下 lunar 因子更一般性猜想的一个特例。我们通过从集合求和推广到多重集求和，解决了这第三个猜想。
英文摘要：
Let $A, B \subseteq \mathbb{N}$ be two finite sets of natural numbers. We say that $B$ is an additive divisor for $A$ if there exists some $C \subseteq \mathbb{N}$ with $A = B+C$. We prove that among those subsets of $\{0, 1, \ldots, k\}$ which have $0$ as an element, the full interval $\{0, 1, \ldots,k\}$ has the most divisors. To generalize to sets which do not have $0$ as an element, we prove a correspondence between additive divisors and lunar multiplication, introduced by Appelgate, LeBrun and Sloane (2011) in their study of a kind of min/max arithmetic. The number of binary lunar divisors is related to compositions of integers which are restricted in that the first part is greater or equal to all other parts. We establish some bounds on such compositions to show that $\{1, \ldots, k\}$ has the most divisors among all subsets of $\{0, 1, \ldots, k\}$. These results resolve two conjectures of LeBrun et al. regarding the maximal number of lunar binary divisors, a special case of a more general conjecture about lunar divisors in arbitrary bases. We resolve this third conjecture by generalizing from sum-sets to sum-multisets.
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