  where b is (f(a1),…,f(an)).  If we have ‘⇔’ in place of ‘⇒’ in the quoted condition, we say that f is an embedding of A into B. Since the language includes =, an embedding of A into B is always one-to-one, though it need not be onto the domain of B. If it is onto, then the inverse map from dom(B) to dom(A) is also a homomorphism, and both the embedding and its inverse are said to be isomorphisms. We say that two structures are isomorphic if there is an isomorphism from one to the other. Isomorphism is an equivalence relation on the class of all structures of a fixed signature K. If two structures are isomorphic then they share all model-theoretic properties; in particular they are elementarily equivalent. 