Differences of Functions with the Same Value Multiset

In a recent article, Ullman and Velleman studied functions a from an abelian group G to itself that can be expressed as a difference of two bijections b, c from G to itself. In this work we relax the condition that b and c are bijections and instead study functions that can be expressed as the difference of two functions with the same value multiset. We construct all possible functions b, c with same value multiset, such that \(a = b - c\) . As a consequence, we obtain a stronger version of Hall’s theorem, which gives a description of b and c in terms of a. We conclude by presenting further directions and questions that relate this new approach to applications of Hall’s theorem.