Varieties, Free Algebras and Equational Logic
Birkhoff's HSP Theorem
The central theorem of universal algebra: a class of algebras is definable by identities exactly when it is closed under homomorphic images, subalgebras and direct products.
Learning objectives
- State the theorem and both directions of its proof
- Follow the free-algebra argument for the hard direction
- Apply the theorem to decide equational definability
The statement
A class K of algebras of a fixed type is an equational class if and only if it is a variety — that is, closed under H, S and P.
The theorem identifies a syntactic notion (definability by equations) with a structural one (closure under three constructions). That identification is what makes universal algebra a subject rather than a collection of analogies.
The easy direction
Every equational class is a variety. Each of the three closures was established on the previous page: homomorphisms preserve term operations; subalgebras inherit them by restriction; products compute them coordinatewise. So an identity holding throughout K holds throughout H(K), S(K) and P(K).
The hard direction
Suppose K is closed under H, S and P. The claim is that K = M(Id(K)) — every algebra satisfying all identities of K already lies in K.
- Let A satisfy every identity of K. Choose a generating set and take X of the same cardinality, so there is a surjection X → A.
- The free algebra FK(X) lies in SP(K), hence in K since K is closed under S and P.
- The surjection X → A extends to a homomorphism FK(X) → A. It is surjective because X generates A.
- Well-definedness of that extension is exactly the hypothesis that A satisfies all identities of K: terms identified in the free algebra must be identified in A.
- So A is a homomorphic image of a member of K, and closure under H gives A ∈ K.
Using the theorem
The practical value is a decision procedure of sorts: to show a class is not equationally definable, exhibit a failure of one closure.
| Class | Fails | Witness |
|---|---|---|
| Fields | P | A product of two fields has zero divisors |
| Finite groups | P | An infinite product of finite groups is infinite |
| Torsion-free abelian groups | H | Z maps onto Z/nZ |
| Simple groups | S, P | Subgroups and products of simple groups need not be simple |
| Cyclic groups | P | A product of cyclic groups need not be cyclic |
| Cancellative semigroups | H | A quotient can lose cancellation |
Birkhoff's theorem guarantees that a variety has an equational basis but gives no way to find one, and no bound on its size. Whether a finite basis exists is a separate and much harder question — the subject of Chapter V §4.
Consequences
- Varieties are exactly the equational classes, so the two terms are used interchangeably from here on.
- The lattice of subvarieties is dual to the lattice of equational theories. This duality is the content of the Galois connection.
- Free algebras exist in every variety, since a variety is closed under S and P.
- Equational logic is complete for varieties — the syntactic and semantic consequence relations agree, as Chapter II §14 establishes.
Frequently asked questions
Does Birkhoff's theorem hold for infinitary algebras?
Not in the same form. The proof uses free algebras whose existence depends on the finitary construction, and the theorem's statement requires modification in the infinitary setting.
Is there an analogous theorem for quasivarieties?
Yes — a class is definable by quasi-identities exactly when it is closed under isomorphism, subalgebras, products and ultraproducts. This is due to Mal'cev and is the natural companion result.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section II.11, book pages 79-84.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.