Variety (universal Algebra) - 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.