Variety (universal algebra) Guide, Meaning , Facts, Information and Description

In universal algebra, a variety is a class of algebraic structures of the same signature satisfying a set of equations.

According to Birkhoff's theorem, a variety (in the sense of Birkhoff) is the same thing as an equational class, namely the kind of variety mentioned in the introduction.

That is, 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.

On the other hand, variety means some collection of algebras for Σ, closed under the three operations:

• take subalgebras
• take homomorphic images
• take cartesian products.

It is simple to see that an equational class satisfies these conditions, so that the burden of Birkhoff's theorem is the converse: classes of algebras that satisfy those conditions must be equational.

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