free algebra

A free algebra is the most generic algebra you can build on a given set of generators: it imposes no relations beyond those forced by the axioms of the class, so its elements are ‘all the things you could possibly write down’ and nothing is accidentally equal. Because it assumes nothing extra, it maps onto every other algebra of the same type — pick where the generators should go, and the rest of the map is forced.

Formally, fix a variety V and a set X of generators. The free algebra F_V(X) on X in V comes with an insertion map X -> F_V(X) satisfying this universal property: for every algebra A in V and every function g: X -> A, there is a unique homomorphism h: F_V(X) -> A extending g. Concretely F_V(X) is the term algebra over X modulo the congruence identifying two terms exactly when the identities of V force them equal; for V = all Σ-algebras (no identities) this congruence is trivial and the free algebra is the term algebra itself.

Free algebras are the engine of universal algebra: every algebra in V is a quotient of some free algebra, identities of V are precisely the term-pairs equal in the free algebras, and F_V(X) on a countably infinite X determines V. Familiar instances abound — the free group on a set, the free monoid (words under concatenation), the free commutative ring Z[x_1, x_2, ...] (the polynomial ring), the free Boolean algebra, the free vector space (with basis X).