Free on a set
You have already met free objects without the name: the polynomial ring R[x] is the free commutative R-algebra on {x}; the free group on a set is the free object among groups. Universal algebra packages all of these. Fix a signature and a class K of algebras (say, all groups, or all lattices). The free algebra F_K(X) on a set X of generators is the algebra in K with a map X → F_K(X) such that any map from X into any A in K extends *uniquely* to a homomorphism F_K(X) → A.
This is the universal property: F_K(X) is the most general algebra in K generated by X, with no relations beyond those forced by the identities of K. The construction is concrete. Start with the term algebra T(X) of all terms in variables X — the *absolutely* free algebra, satisfying no identities at all. Then quotient by the smallest congruence that forces all the defining identities of K. The result is F_K(X), and its elements are terms-up-to-provable-equality.
Free group on X = {x, y}, written F(x,y).
Elements = reduced words in x, y, x^{-1}, y^{-1} (no adjacent a a^{-1}).
From term algebra to free group:
term: ·( ·(x, inv(x)), y )
group identities force ·(x, inv(x)) approx e and ·(e, y) approx y
so this term collapses to: y
Two terms land in the same class of F(x,y) iff group axioms PROVE them equal.
Universal property in action. To map F(x,y) -> S_3, just choose images:
x |-> (1 2), y |-> (1 2 3)
This extends to a UNIQUE homomorphism phi: F(x,y) -> S_3, with e.g.
phi(x y x^{-1}) = (1 2)(1 2 3)(1 2) = (1 3 2).
Every such choice of two elements of S_3 gives exactly one homomorphism
-- that is precisely what 'free on {x,y}' means.Identities as equalities in the free algebra
Here is the payoff that makes free algebras the engine of the whole theory. An identity s ≈ t in variables x₁,…,xₙ holds in *every* algebra of class K iff s and t name the same element of the free algebra F_K(x₁,…,xₙ). Checking an identity on infinitely many algebras reduces to a single equality in one algebra. This is why F_K(X) is sometimes called the algebra of free terms: it is the universal place to test equations.
Three operations on classes: H, S, P
Given a class K of algebras of one signature, three operators produce new classes. H(K) is all homomorphic images of members of K; S(K) is all subalgebras; P(K) is all (possibly infinite) direct products. Each is justified by a closure fact you can check by hand: a quotient, a subalgebra, and a product of algebras satisfying an identity all satisfy that same identity. These three are the load-bearing operations of the next guide.
- H — if A satisfies s ≈ t and A ↠ B, then B satisfies s ≈ t: evaluate the identity in B by lifting to A through the surjection.
- S — a subalgebra evaluates terms using the very same operations restricted, so any identity true in A is true in every subalgebra.
- P — operations in a product act coordinatewise, so an identity holds in ∏ Aᵢ iff it holds in each Aᵢ; closure under P is immediate.