 This implies not only that every nonempty rectangular band is isomorphic to one coming from a pair of sets, but also these sets are uniquely determined up to a canonical isomorphism, and all homomorphisms between bands come from pairs of functions between sets.If the set "I" is empty in the above result, the rectangular band "I" × "J" is independent of "J", and vice versa.This is why the above result only gives an equivalence between nonempty rectangular bands and pairs of nonempty sets.