The weak acyclic matching property in abelian groups

A matching from a finite subset $A\subset {\mathbb{Z}}^n$ to another subset $B\subset {\mathbb{Z}}^n$ is a bijection $f:A\to B$ with the property that $a+f(a)$ never lies in $A$. A matching is called acyclic if it is uniquely determined by its multiplicity function. Alon et al. established the acyclic matching property for ${\mathbb{Z}}^n$, which was later extended to all abelian torsion-free groups. In a prior work, the authors of this paper settled the acyclic matching property for all abelian groups. The objective of this note is to explore a related concept, known as the weak acyclic matching property, within the context of abelian groups.