Given finite sets $S$ and $T$, you have a commutative function $\bullet: S\times S \rightarrow T$.
What do you call the following property?: Whenever there are $x,y,z$ so that $x\bullet z = y \bullet z$ then for all $w$, we have $x\bullet w = y\bullet w$. (It's a kind of agreement property vaguely similar to cancellation.)
The motivating context: I'm looking for situations where $\bullet$ can be mapped onto addition on integers, i.e. where there is a mapping $f:S\rightarrow \mathbb{N}$ and $g:T\rightarrow \mathbb{N}$ such that $f(x)+f(y) = g(x\bullet y)$ for all $x,y\in S$. I know that if the operator lacks the above property, there can be no such homomorphism*. But is the property sufficient, or what's missing?