In 2011, Nathanson proposed several questions on minimal complements in a group or a semigroup. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

2011 年，Nathanson 提出了几个关于群或半群中的极小补的问题。最小补和作为最小补的概念导致了共同最小对的概念，这在作者的先前工作中被考虑过。在本文中，我们研究整数和更高秩的自由阿贝尔群中的哪种类型的子集可以成为共同极小对的一部分。我们表明大多数空缺序列具有此属性。根据所建立的条件，可以证明任何有限生成的阿贝尔群的任何无限子集都具有不可数多的子集，这些子集是共同极小对的一部分。此外，可以选择不可数集合的集合，以便它们满足某些代数性质。