Let and be two algebras over the same signature , with carriers and , respectively (cf. universal algebra). is a subalgebra of if and every function of is the restriction of the respective function of on .
The (direct) product of algebras and is an algebra whose carrier is the Cartesian product of and and such that for every and all and all ,
A nonempty class of algebras over the same signature is called a variety if it is closed under subalgebras, homomorphic images, and Cartesian products over arbitrary families of structures belonging to the class.
A class of algebras is said to satisfy the identity if this identity holds in every algebra from this class. Let be a set of identities over signature . A class of algebras over is called an equational class if it is the class of algebras satisfying all identities from . In this case, is said to be axiomatized by .
Birkhoff's theorem states that is an equational class iff it is a variety.