Free lattice

Summary

The free lattice on a set $X$ is the lattice generated by $X$ in which no relations hold except those forced by the lattice laws. It has a universal property: every map from $X$ into a lattice extends uniquely to a lattice homomorphism from the free lattice. Its elements can be described by lattice terms in the letters of $X$. Deciding when two terms are equal is the word problem for free lattices. The free lattice on two generators has four elements, while the free lattice on three generators is already infinite.