According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products.
An equational law is a pair of such words; the axiom consisting of the words v and w is written as v = w. A theory consists of a signature, a set of variables, and a set of equational laws.
A sufficient defining equation is the associative law: The class of groups forms a variety of algebras of signature (2,0,1), the three operations being respectively multiplication (binary), identity (nullary, a constant) and inversion (unary).
To express the scalar multiplication with elements from R, we need one unary operation for each element of R. If the ring is infinite, we will thus have infinitely many operations, which is allowed by the definition of an algebraic structure in universal algebra.
We will then also need infinitely many identities to express the module axioms, which is allowed by the definition of a variety of algebras.
The fields do not form a variety of algebras; the requirement that all non-zero elements be invertible cannot be expressed as a universally satisfied identity (see below).
Given a class of algebraic structures of the same signature, we can define the notions of homomorphism, subalgebra, and product.
Garrett Birkhoff proved that a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and arbitrary products.
One direction of the equivalence mentioned above, namely that a class of algebras satisfying some set of identities must be closed under the HSP operations, follows immediately from the definitions.
Besides varieties, category theorists use two other frameworks that are equivalent in terms of the kinds of algebras they describe: finitary monads and Lawvere theories.
This is a more general notion than "finitary algebraic category" because it admits such categories as CABA (complete atomic Boolean algebras) and CSLat (complete semilattices) whose signatures include infinitary operations.
In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded arity.
A pseudovariety is usually defined to be a class of algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products.
For pseudovarieties, there is no general finitary counterpart to Birkhoff's theorem, but in many cases the introduction of a more complex notion of equations allows similar results to be derived.
[2] Pseudovarieties are of particular importance in the study of finite semigroups and hence in formal language theory.