Free Algebra

An algebra in a given signature and variety generated by a set of free generators with no defining relations other than the identities required by the signature and the variety; equivalently, an object F(X) equipped with an inclusion of the generator set X such that every function from X into any algebra A of the same variety extends uniquely to a homomorphism F(X) → A.

An algebra in a variety generated freely by a set of generators; it satisfies the universal mapping property that every function from the generator set to any algebra of the variety extends uniquely to a homomorphism from the free algebra.