Variety (universal algebra)

In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. 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. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.

A covariety is the class of all coalgebraic structures of a given signature.

A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial equations. They are formally quite distinct and their theories have little in common.

The term "variety of algebras" refers to algebras in the general sense of universal algebra; there is also a more specific sense of algebra, namely as algebra over a field, i.e. a vector space equipped with a bilinear multiplication.

The class of all semigroups forms a variety of algebras of signature (2), meaning that a semigroup has a single binary operation. 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). The familiar axioms of associativity, identity and inverse form one suitable set of identities:

The class of rings also forms a variety of algebras. The signature here is (2,2,0,0,1) (two binary operations, two constants, and one unary operation).

If we fix a specific ring R, we can consider the class of left R-modules. 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. So the left R-modules do form 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.[citation needed]

The cancellative semigroups also do not form a variety of algebras, since the cancellation property is not an equation, it is an implication that is not equivalent to any set of equations. However, they do form a quasivariety as the implication defining the cancellation property is an example of a quasi-identity.

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.[1] This is 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 operations of homomorphism, subalgebra, and product.

The class of algebras satisfying some set of identities will be closed under the HSP operations. Proving the converse—classes of algebras closed under the HSP operations must be equational—is more difficult.

Using Birkhoff's theorem, we can for example verify the claim made above, that the field axioms are not expressable by any possible set of identities: the product of fields is not a field, so fields do not form a variety.

A subvariety of a variety of algebras V is a subclass of V that has the same signature as V and is itself a variety, i.e., is defined by a set of identities.

Viewing a variety V and its homomorphisms as a category, a subvariety U of V is a full subcategory of V, meaning that for any objects a, b in U, the homomorphisms from a to b in U are exactly those from a to b in V.

This generalizes the notions of free group, free abelian group, free algebra, free module etc. It has the consequence that every algebra in a variety is a homomorphic image of a free algebra.

Since varieties are closed under arbitrary direct products, all non-trivial varieties contain infinite algebras. Attempts have been made to develop a finitary analogue of the theory of varieties. This led, e.g., to the notion of variety of finite semigroups. This kind of variety uses only finitary products. However, it uses a more general kind of identities.

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. Not every author assumes that all algebras of a pseudovariety are finite; if this is the case, one sometimes talks of a variety of finite algebras. 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. Eilenberg's theorem, often referred to as the variety theorem, describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.