On Co-Minimal Pairs in Abelian Groups

Abstract

A pair of non-empty subsets (W,W′) in an abelian group G is a complement pair if W+W′=G. W′ is said to be minimal to W if W+(W′∖{w′})≠G,∀w′∈W′. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs (W,W′) such that both W and W′ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in Z, also holds in any abelian group.