Let and be two algebras over the same signature , with carriers A and B, respectively (cf. universal algebra). is a subalgebra of if and every function of is the restriction of the respective function of on B.
The (direct) product of algebras and is an algebra whose carrier is the Cartesian product of A and B and such that for every and all and all ,

A nonempty class K of algebras over the same signature is called a variety if it is closed under subalgebras, homomorphic images, and direct products.

A class of algebras is said to satisfy the identity s = t if this identity holds in every algebra from this class. Let E be a set of identities over signature . A class K of algebras over is called an equational class if it is the class of algebras satisfying all identities from E. In this case, K is said to be axiomatized by E.

Birkhoff's theorem states that K is an equational class iff it is a variety.