Terms, Free Algebras and Equational Logic

Identities and Birkhoff's HSP Theorem

Closed under homomorphic images, subalgebras and products — that and nothing more is what it takes to be defined by equations. Birkhoff's theorem is a completeness result in disguise.

Engineering · Mathematics5 min readKV-MATH-0221

Learning objectives

State what it means for an algebra to satisfy an identity.
Verify that equational classes are closed under H, S and P.
Prove the converse: HSP-closed classes are equational.
Present M and Id as a Galois connection and read off the closure operators.
Apply the theorem to show a given class is or is not a variety.
Explain the significance of the theorem as a syntax/semantics bridge.

01Satisfaction

An algebra A satisfies the identity p ≈ q when the induced term operations coincide: p^A(a⃗) = q^A(a⃗) for every assignment of elements to the variables.

A ⊨ p ≈ q ⟺ p^A = q^A as functions

Identities are implicitly universally quantified. A class K satisfies p ≈ q when every member does, written K ⊨ p ≈ q.

Two derived notations do most of the work: M(Σ) is the class of all algebras satisfying every identity in Σ, and Id(K) is the set of all identities satisfied by every member of K.

02The easy direction

Equationally defined classes are closed under all three operators, by direct verification.

Subalgebras
If p^A = q^A on all of A, the restriction to a subuniverse still agrees, because term operations restrict. So M(Σ) is S-closed.
Homomorphic images
Apply α to p^A(a⃗) = q^A(a⃗) and use that α commutes with term operations. Surjectivity ensures every tuple of the image is covered.
Direct products
Term operations in a product are computed coordinatewise, so agreement in every factor gives agreement in the product.
Conclusion
M(Σ) is a variety for any Σ. The content of the theorem is entirely in the converse.

03The converse

ProcedureEvery HSP-closed class is equationally defined

in: HSP-closed K → out: K = M(Σ) for Σ = Id(K)

input: class K with H(K) = S(K) = P(K) = K
let Σ := Id(K), the identities holding throughout K
clearly K ⊆ M(Σ); show the reverse inclusion
take A ∈ M(Σ); choose a set X and a surjection α : X → A
α extends to a surjective homomorphism β : T(X) → A
the free algebra F_K(X) = T(X)/θ_K(X) lies in SP(K) ⊆ K
since A ⊨ Σ, we have θ_K(X) ⊆ ker(β)
so β factors through F_K(X), giving a surjection F_K(X) → A
therefore A ∈ H(K) = K
output: K = M(Id(K)), so K is equationally defined

The load-bearing steps are F_K(X) ∈ SP(K) — proved on the free algebras page — and the inclusion θ_K(X) ⊆ ker(β), which is exactly the assumption that A satisfies K's identities. Caveat: X must be large enough to surject onto A, so the argument uses free algebras of arbitrary rank, not just countable ones.

Key resultBirkhoff's HSP theorem

A class of algebras of a fixed type is definable by a set of identities if and only if it is closed under homomorphic images, subalgebras and direct products. Equivalently: the varieties are exactly the equational classes, and V(K) = HSP(K) = M(Id(K)).

04The Galois connection view

M and Id form an antitone Galois connection between classes of algebras and sets of identities, and the theorem identifies the closed sets on both sides.

Algebra side

Closed classes are varieties

K ↦ M(Id(K)) is the closure operator, and it equals HSP. Its closed classes are the varieties.

Identity side

Closed sets are equational theories

Σ ↦ Id(M(Σ)) is deductive closure. Its closed sets are the equational theories, characterised as fully invariant congruences on the term algebra.

The two lattices of closed sets are dually isomorphic. The lattice of varieties of a fixed type is therefore the order dual of the lattice of equational theories — a correspondence used constantly when studying the structure of varietal lattices.

05Applying the theorem

Is the class a variety?
Class	Variety?	Reason
Groups	Yes	Equational in type (2,1,0).
Abelian groups	Yes	Add commutativity.
Lattices	Yes	The eight lattice identities.
Boolean algebras	Yes	Equational in type (2,2,1,0,0).
Fields	No	Not closed under P — a product of fields has zero divisors.
Torsion-free abelian groups	No	Not closed under H — quotients have torsion.
Simple groups	No	Not closed under S or P.
Finite groups	No	Not closed under infinite P.
Cancellative semigroups	No	Not closed under H.

The negative cases are the instructive ones. Each fails a specific closure property, and identifying which one immediately tells you no set of identities can define the class — no search for axioms is needed.

06Why the theorem matters

Bridge

Syntax meets semantics

A purely syntactic notion (definable by equations) coincides exactly with a purely algebraic one (closed under three constructions). Completeness results of this shape are rare and valuable.

Method

Two ways to specify a variety

Either list identities or list generators and close under HSP. Both give the same class, so one may switch to whichever is convenient for the argument at hand.

Programme

Varieties become the objects

Because the notion is robust, the lattice of varieties of a given type becomes a legitimate object of study, and classifying it is the central programme of the subject.

NoteWhat the theorem does not give

It does not say the defining set of identities is finite. Whether a finitely generated variety has a finite equational basis is the finite basis problem, which is genuinely hard and is addressed in the Model-Theoretic and Research Frontier streams.

Frequently asked

Does HSP need to be applied in that order?

Yes. The inclusions SH ≤ HS, PH ≤ HP and PS ≤ SP all push H left and P right, so HSP absorbs any composite. Other orderings are not idempotent and do not produce the variety generated.

Is the set of defining identities unique?

No — many different sets define the same variety. What is unique is the deductive closure Id(M(Σ)), the equational theory. Two sets define the same variety exactly when they have the same closure, which is why equational theories rather than axiom sets are the canonical objects.

Does the theorem hold for quasi-identities?

There is an analogue: classes definable by quasi-identities — implications between conjunctions of equations — are exactly those closed under S, P and ultraproducts, and containing a trivial algebra. Dropping H is what distinguishes quasivarieties from varieties, and the result requires ultraproducts, which is why it belongs to the model-theoretic part of the subject.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.