关于非极小补
On non-minimal complements
摘要：
最小补数的概念是由Nathanson在2011年提出的。从那时起，集合的极小补的存在性或不存在性得到了广泛的研究。近年来，反问题的研究，即，当最小补充获得牵引力时，这些集合可以或不可以发生。例如，Kwon，Alon-Kravitz-Larson，Burcroff-Luntzlara的作品以及作者的作品，阐明了这方面的一些问题。这些作品主要集中在该组的整数，或对阿贝尔群。在这项工作中，我们的动机是双重的：（i）显示一些新的结果的反问题，（ii）集中在反问题不一定阿贝尔群。作为一个副产品，我们得到了新的结果，非极小补的整数组，更一般地说，在任何非极小生成的阿贝尔群的正秩和任何自由阿尔贝群的正秩。此外，我们还证明了在这些群中存在不可数多个子集，它们是“鲁棒”的非极小补集。
英文摘要：
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.
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