FD(2) — the free distributive lattice on two generators

The free distributive lattice on two generators $x, y$ consists of all distinct lattice terms one can build from $\{x, y\}$ using $\wedge$ and $\vee$ subject only to the distributive-lattice axioms. The answer is small: four elements $\{x \wedge y,\; x,\; y,\; x \vee y\}$, forming a four-element Boolean lattice $B_2$ — the “kite” or $2 \times 2$ grid.

This is the universal property at work: for any distributive lattice $L$ and any two elements $u, v \in L$, there is a unique homomorphism $FD(2) \to L$ sending $x \mapsto u$ and $y \mapsto v$.

Where it appears

• As $B_2$, it is also the powerset of a 2-element set.
• The number $|FD(n)|$ is the Dedekind number $D(n)$, famously difficult to compute: $D(2) = 4$, $D(3) = 18$, $D(4) = 166$, $D(5) = 7\,828\,354$, and $D(9)$ was only determined in 2023.

The smallness of $FD(2)$ relative to the explosion at higher $n$ is a quiet reminder that distributivity is permissive enough to allow rich structure as soon as the generators become more than a pair.