Free lattice

Remarks

A free lattice on $X$ exists and is unique up to a unique isomorphism compatible with the maps from $X$; it is written $\mathrm{FL}(X)$. One construction takes the set of lattice terms built from the letters of $X$ with $\vee$ and $\wedge$. It identifies two terms $s$ and $t$ when the identity $s = t$ holds in every lattice. The map $\iota$ is injective, and $\mathrm{FL}(X)$ is generated by $\iota(X)$ as a lattice. Example: for $X = \set{x, y}$, $\mathrm{FL}(X) = \set{x, y, x \wedge y, x \vee y}$ with $x \wedge y < x, y < x \vee y$. Every lattice term in $x$ and $y$ reduces to one of these four by the lattice laws. These four are distinct because they are distinct in the lattice of subsets of $\set{1, 2}$ under $x \mapsto \set{1}$, $y \mapsto \set{2}$. Non-example: the two-element chain $\set{x, y}$ with $x < y$, with $\iota$ the identity, is not free on $\set{x, y}$. The map $g$ with $g(x) = \set{1}$ and $g(y) = \set{2}$ into the subsets of $\set{1, 2}$ has no extension $h$, since a lattice homomorphism is monotone and $\set{1} \not\subseteq \set{2}$. For $\abs{X} \ge 3$, the free lattice $\mathrm{FL}(X)$ is infinite.