Terms, Free Algebras and Equational Logic

Free Algebras and the Universal Mapping Property

A free algebra imposes exactly the equalities the class demands and no others. Build it as a quotient of the term algebra, and Birkhoff's theorem falls out.

Engineering · Mathematics5 min readKV-MATH-0220

Learning objectives

State the universal mapping property and derive uniqueness from it.
Construct F_K(X) as T(X)/θ_K(X).
Prove that F_K(X) lies in SP(K), hence in V(K).
Explain why free algebras exist for every non-trivial class.
Use free algebras to reduce identity-checking to a single algebra.
Recognise free algebras in familiar varieties.

01The universal mapping property

An algebra F with a map ι : X → F has the universal mapping property for a class K over X when every map from X into a member of K extends uniquely to a homomorphism from F.

∀ A ∈ K, ∀ α : X → A, ∃! β : F → A with β ∘ ι = α

Uniqueness forces ι(X) to generate F: otherwise two homomorphisms could agree on ι(X) and differ elsewhere.

Key resultThe property determines the algebra up to isomorphism

Any two algebras with the universal mapping property for the same K and X are isomorphic, by the standard argument: each factors through the other, and the composites are endomorphisms fixing the generators, hence identities by uniqueness. So 'the' free algebra is well defined.

02Construction as a quotient

ProcedureBuilding F_K(X)

in: K, X → out: free algebra F_K(X)

input: class K of algebras of type F, variable set X
form the term algebra T(X)
θ_K(X) := { ⟨p, q⟩ ∈ T(X)² : K ⊨ p ≈ q }
i.e. p and q induce the same operation in every member of K
verify θ_K(X) ∈ Con T(X) (it is an intersection of kernels)
F_K(X) := T(X)/θ_K(X)
ι : X → F_K(X) sends x to its class x/θ_K(X)
output: F_K(X) with the universal mapping property for K over X

θ_K(X) is the intersection of the kernels of all homomorphisms from T(X) into members of K, hence a congruence. Caveat: ι need not be injective if K contains only trivial algebras — then F_K(X) is trivial and distinct variables collapse.

The construction says: start with no equalities at all, then impose exactly those that K forces. That is the precise sense in which the free algebra is as unconstrained as membership in K permits.

03Free algebras belong to the variety

The step that makes Birkhoff's theorem work is that F_K(X) is not merely related to K but lies in the variety K generates — indeed in SP(K), which is stronger.

θ_K(X) is an intersection of kernels
For each A ∈ K and each map α : X → A, the induced homomorphism T(X) → A has a kernel. θ_K(X) is the intersection of all of them.
Intersecting kernels gives a subdirect embedding
T(X)/⋂ker(β_i) embeds in the product of the T(X)/ker(β_i), each of which embeds in the corresponding A ∈ K.
So F_K(X) ∈ SP(K)
A subalgebra of a product of members of K. In particular F_K(X) ∈ V(K), with no use of H.
Consequence
F_K(X) satisfies exactly the identities of K, and being in the variety, it is a legitimate test object for them.

NoteWhy this is the crux of HSP

To prove a HSP-closed class K is equationally defined, one shows any algebra satisfying K's identities is a homomorphic image of a free algebra F_K(X) for suitable X. Since F_K(X) ∈ K by the above, and K is closed under H, the algebra lies in K. The whole theorem turns on F_K(X) being a member.

04Free algebras as universal test objects

An identity holds throughout a class exactly when it holds in the free algebra on countably many generators. This collapses a statement about a proper class into a statement about one algebra.

K ⊨ p ≈ q ⟺ F_K(X) ⊨ p ≈ q for |X| ≥ the number of variables in p, q

Countably many generators suffice for all identities at once, since each identity involves finitely many variables.

This is why F_V(ω), the free algebra on a countably infinite generating set, is such a central object. Its congruence lattice, its subalgebras and its endomorphism monoid all encode information about the entire variety.

05Free algebras in familiar varieties

What freeness produces
Variety	Free algebra on X	Note
Sets (empty type)	X itself	No operations, nothing to impose.
Semigroups	non-empty words over X	Free semigroup; concatenation.
Monoids	all words over X including empty	Free monoid.
Groups	reduced words over X ∪ X⁻¹	Free group; normal form by reduction.
Abelian groups	free ℤ-module on X	Direct sum of copies of ℤ.
Commutative rings with 1	ℤ[X], polynomials	The universal property of polynomial rings.
Boolean algebras	finite: 2^(2^\|X\|) elements	Free Boolean algebra on n generators has 2^(2ⁿ) elements.
Lattices	free lattice on X	Infinite for \|X\| ≥ 3; word problem solvable.
Distributive lattices	finite for finite X	Much smaller than free lattices.

The polynomial ring example is worth pausing on: ℤ[x₁,…,xₙ] is exactly the free commutative ring with unit on n generators, and the familiar universal property of polynomial rings is a special case of the universal mapping property stated here.

06Free spectra and growth

The function sending n to |F_V(n)| is the free spectrum of the variety, and its growth rate is a genuine invariant.

Doubly exponential

Boolean algebras

|F(n)| = 2^(2ⁿ). The free algebra records every Boolean function of n variables, so growth is as fast as it can be for a locally finite variety.

Polynomial or linear

Vector spaces over a fixed finite field

|F(n)| = qⁿ. Very slow growth, reflecting how few term operations exist.

CautionFree spectra are computed values, not durable facts

Specific free-spectrum numbers for named varieties depend on the presentation, on characteristic and, for finitely generated varieties, on which generating algebra was used. They are catalogue data. Where a concrete value is needed, compute it with UACalc or a computer algebra system rather than quoting a figure from a text — see the sourcing policy page.

Frequently asked

Does a free algebra exist over any class?

Yes, by the quotient construction, though it may be trivial. If every member of K is a one-element algebra then θ_K(X) is everything and F_K(X) is trivial. For any class containing an algebra with at least two elements, the free algebra is non-trivial and ι is injective.

Is F_K(X) the same as F_{V(K)}(X)?

Yes. A class and the variety it generates satisfy exactly the same identities, so θ_K(X) = θ_{V(K)}(X) and the free algebras coincide. This is convenient: one may compute the free algebra from a small generating class rather than the whole variety.

Why does the free algebra need only countably many generators?

Because every identity involves only finitely many variables. To test all identities at once, countably many generators suffice. Free algebras on larger generating sets exist and matter for other purposes — representing large members of the variety — but not for deciding identities.

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.