Variety (universal algebra)

In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic images, subalgebras and cartesian products.

A variety of algebras should not be confused with an algebraic variety. Intuitively, a variety of algebras is an equationally defined collection of algebras, while an algebraic variety is an equationally defined collection of elements from a single algebra. The two are named alike by analogy, but they are formally quite distinct and their theories have little in common.

Birkhoff's theorem

The equivalence of the two definitions given above is of fundamental importance in universal algebra. It was proved by Garrett Birkhoff, and is generally known as Birkhoff's theorem, or as the HSP theorem (H, S and P standing respectively for the closure operations of homomorphism, subalgebra and product).

Formally, suppose we fix a signature Σ. An equational class for Σ is the set of all models, in the sense of model theory for example, that satisfy equations in a given set E. Those equations are statements from the predicate calculus involving universal quantifiers and equality only: each is a mathematical identity enforced in each model, for example the commutative law, or the absorption law.

It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations, so the burden of Birkhoff's theorem is the converse: classes of algebras that satisfy those conditions must be equational.

Examples

The class of all semigroups forms a variety of algebras of signature (2). A sufficient defining equation is the associative law:

x(yz) = (xy)z

It satisfies the HSP closure requirement, since any subset closed under multiplication, any homomorphic image and any direct product of semigroups is also a semigroup.

The class of groups forms a class of algebras of signature (2,1,0), the three operations being respectively multiplication, inversion and identity. Any subset of a group closed under multiplication, under inversion and under identity (ie. containing the identity) forms a subgroup. Likewise, the collection of groups is closed under homomorphic image and under direct product. Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities. In fact, the familiar axioms of associativity, inverse and identity form one suitable set of identities:

x(yz) = (xy)z

1 x = x 1 = x

x x^-1 = x^{-1 x = 1}

Notice that although every group is a semigroup, the class of groups does not form a subvariety of the variety of semigroups. This is because not every subsemigroup of a group is a group.

The class of abelian groups, considered again with signature (2,1,0), also has the HSP closure properties. It forms a subvariety of the variety of groups, and can be defined equationally by the three group axioms above together with the commutativity law:

xy = yx

Finitary analogues

Since varieties are closed under arbitrary cartesian products, all non-trivial varieties contain infinite algebras. It follows that the theory of varieties is of limited use in the study of finite algebras, where one must often apply techniques particular to the finite case. With this in mind, attempts have been made to develop a finitary analogue of the theory of varieties.

A variety of finite algebras, sometimes called a pseudovariety, is usually defined to be a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. 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.

Pseudovarieties are of particular importance in the study of finite semigroups and hence in formal language theory. Eilenberg's theorem describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.