I am an algebraist, and I am wondering if there is a definition for the following:

Let $A_1$, $A_2$, $\ldots, A_n$ be sets of integers (or more generally, subsets of a group $G$). Say that (for the purposes of this question) $A_1\times A_2\cdots\times A_n$ is *special* provided that whenever $a_1+a_2+\cdots+a_n=b_1+b_2\cdots+b_n$ with each $a_i,b_i\in A_i$, $1\leq i\leq n$, then $a_i=b_i$ for all $i$, $1\leq i\leq n$.

Is there some terminology for this property? Any information would be appreciated.