Free distributive lattice

Summary

The free distributive lattice on a set $S$ is the distributive lattice generated by $S$ with no extra relations. The only equalities in it are those that the lattice laws and distributivity force. Every map from $S$ into a distributive lattice extends to exactly one lattice homomorphism from it. Unlike the free lattice on three or more generators, the free distributive lattice on a finite set is finite. On $n$ generators it has $M(n) - 2$ elements, where $M(n)$ is the Dedekind number.