Through some research I have been doing, I came to realize that the functions I am working on are "Multi-Injective", however I can't find any literature on such functions, and I figure that I might just be using the incorrect term.
By "Multi-Injective", I'm borrowing the structure of the term from "Multilinear", so the function is injective with respect to each individual variable. For example, one could consider a "Tri-Injective" function $f:A^3 \rightarrow B$, such that
$$ f(a,b,c) = f(d,b,c) \implies a = d $$ $$ f(a,b,c) = f(a,d,c) \implies b = d $$ $$ f(a,b,c) = f(a,b,d) \implies c = d $$
Thanks for any help!