论文标题
关于非最小补充
On non-minimal complements
论文作者
论文摘要
Nathanson在2011年引入了最小补充的概念。从那时起，已经对集合的最小补充的存在或不存在。最近，对逆问题的研究，即，随着最小补充的吸引力，该集合可能会或不能发生。例如，夸文（Kwon），阿隆（Alon） - 克拉维兹（Kravitz）的作品 - 拉尔森（Larson），伯克罗夫（Burcroff） - 兰兹拉拉（Luntzlara）以及作者的作品，阐明了朝这个方向上的一些问题。这些作品主要集中在整数或阿贝尔群体上。在这项工作中，我们的动机是两个方面：（i）在反问题上显示一些新结果，（ii）专注于不一定是阿贝尔群体的反问题。作为副产品，我们在整数组中获得了非最小补充的新结果，更普遍地，在任何有限生成的Abelian积极等级和任何自由的Abelian Abelian积极等级组中，我们都会获得新的结果。此外，我们在此类群体中表明了许多非最小值补充中许多子集的存在。
The notion of minimal complements was introduced by Nathanson in 2011. Since then, the existence or the inexistence of minimal complements of sets have been extensively studied. Recently, the study of inverse problems, i.e., which sets can or cannot occur as minimal complements has gained traction. For example, the works of Kwon, Alon--Kravitz--Larson, Burcroff--Luntzlara and also that of the authors, shed light on some of the questions in this direction. These works have focussed mainly on the group of integers, or on abelian groups. In this work, our motivation is two-fold: (i) to show some new results on the inverse problem, (ii) to concentrate on the inverse problem in not necessarily abelian groups. As a by-product, we obtain new results on non-minimal complements in the group of integers and more generally, in any finitely generated abelian group of positive rank and in any free abelian group of positive rank. Moreover, we show the existence of uncountably many subsets in such groups which are "robust" non-minimal complements.
