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.

A signature (in this context) is a set, whose elements are called operations, each of which is assigned a natural number (0, 1, 2, ...) called its arity. Given a signature σ and a set V, whose elements are called variables, a word is a finite rooted tree in which each node is labelled by either a variable or an operation, such that every node labelled by a variable has no branches away from the root and every node labelled by an operation o has as many branches away from the root as the arity of o. 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. Any theory gives a variety of algebras as follows. Given a theory T, an algebra of T consists of a set A together with, for each operation o of T with arity n, a function o_A : Aⁿ → A such that for each axiom v = w and each assignment of elements of A to the variables in that axiom, the equation holds that is given by applying the operations to the elements of A as indicated by the trees defining v and w. The class of algebras of a given theory T is called a variety of algebras.

Given two algebras of a theory T, say A and B, a homomorphism is a function f : A → B such that

for every operation o of arity n. Any theory gives a category where the objects are algebras of that theory and the morphisms are homomorphisms.