Birkhoff's HSP theorem

In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature that is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras is usually called a finitary algebraic category.

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. Intuitively, a variety of algebras is an equationally defined collection of algebras, while an algebraic variety is an equationally defined collection of elements from a single algebra. The two are named alike by analogy, but they are formally quite distinct and their theories have little in common.

Garrett Birkhoff proved the equivalence of the two definitions of a 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 universally quantified equations, asserting equality between terms. A model satisfies these equations if they are true in the model for every 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, satisfied by commutative algebras, and the absorption law, satisfied by lattices.

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