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. In the context of category theory, a variety of algebras is usually called a finitary algebraic category.

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

Garrett Birkhoff proved equivalent the two definitions of variety given above, a result of fundamental importance to universal algebra and known as Birkhoff's theorem or as the HSP theorem. H, S, and P stand, respectively, for the closure operations of homomorphism, subalgebra, and product.

An equational class for some signature Σ is the collection of all models, in the sense of model theory, that satisfy some set E of equations, asserting equality between terms. A model satisfies these equations if they are true in the model for any valuation of the variables. The equations in E are then said to be identities of the model. Examples of such identities are the commutative law, characterizing commutative algebras, and the absorption law, characterizing lattices.

It is simple to see that the class of algebras satisfying some set of equations will be closed under the HSP operations. Proving the converse--classes of algebras closed under the HSP operations must be equational--is much harder.

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 homomorphic image, any subset closed under multiplication 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:

$x(yz)=(xy)z$

$1x=x1=x$

$xx^{-1}=x^{-1}x=1$

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:

$xy=yx$

Variety of finite algebras

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, often referred to as the variety theorem describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.

Category theory

If A is a finitary algebraic category, then the forgetful functor

$U:A\to {\mathbf {Set}}$

is monadic. Even more, it is strictly monadic, in that the comparison functor

$K:A\to {\mathbf {Set}}^{\mathbb {T} }$

is an isomorphism (and not just an equivalence). Here, ${\mathbf {Set}}^{\mathbb {T} }$ is the Eilenberg-Moore category on ${\mathbf {Set}}$. In general, one says a category is an algebraic category if it is monadic over ${\mathbf {Set}}$.

References

Two monographs available free online:

• Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.
• Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices, Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8.