Free algebra in a variety

Summary

The free algebra of a variety $\mathcal{V}$ on a set $X$ is the algebra of $\mathcal{V}$ generated by $X$ with as few relations as possible. The only relations among the generators are those forced by the identities of $\mathcal{V}$. Its defining property is that every map from $X$ to an algebra of $\mathcal{V}$ extends to exactly one homomorphism. Free groups, free abelian groups, polynomial rings and free monoids are free algebras of their varieties. Every algebra of $\mathcal{V}$ is a quotient of a free algebra, which makes free algebras the starting point of presentations.