  This construction has some variants. We can define an equivalence relation on the domain of a product C, and then take a structure D whose elements are the equivalence classes; the predicate symbols are interpreted in D so as to make the natural map from dom(C) to dom(D) a homomorphism. In this case the structure D is called a reduced product of the factors of C. It is a reduced power of A if all the factors are equal to A; in this case the diagonal map from A to D is the one got by taking each element a to the equivalence class of the element (a,a,…).