 The structure of varieties as classes of all models of some equational theory is also of great interest. The earliest result in this area is Birkhoff’s theorem that a class of algebras is a variety if and only if it is closed under formation of quotients (homomorphic images), subalgebras, and arbitrary (including empty and infinite) direct products. This “modern algebra” result constitutes a completeness theorem for equational logic in terms of its models. Its elementary counterpart is the theorem that the equational theories on a free algebra \(F(V)\), defined as the deductively closed sets of equations that use variables from \(V\), are exactly its substitutive congruences.