Variety (universal algebra)

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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.

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Terminology

A variety of algebras should not be confused with an algebraic variety,
which means a set of solutions to a system of polynomial equations. The
y 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.