Free Boolean algebra

Summary

The free Boolean algebra on a set $S$ is the Boolean algebra generated by $S$ with no extra relations. The only equalities in it are those that hold in every Boolean algebra. Every map from $S$ into a Boolean algebra extends to exactly one homomorphism from it. On $n$ generators it is the algebra of all Boolean functions of $n$ variables, with $2^{2^n}$ elements. It is also the Lindenbaum–Tarski algebra of classical propositional logic. Its propositional variables are the elements of $S$.