 In the 1930s Garrett Birkhoff (1911–1996) established the fundamental results of equational logic, namely (1) equational classes of algebras are precisely the classes closed under homomorphisms, subalgebras and direct products, and (2) equational logic is based on five rules: reflexivity, symmetry, transitivity, replacement, and substitution. In the 1940s, Tarski joined in this development of equational logic; the subject progressed rapidly from the 1950s till the present time.