---
tag: FMCWXA0WT4
kind: definition
name: Free lattice
msc: [06B25, 08B20]
defines: WP20S9NZ15
uses: [WH4SFAACAS, BNVRR925X8, 6D3WA7N2W4]
sources:
  - url: https://en.wikipedia.org/wiki/Free_lattice
---

## Definition

Let $X$ be a set.
A *free lattice* on $X$ is a [[lattice|WH4SFAACAS]] $F$ together with a map $\iota \colon X \to F$ with the following property.
For every lattice $L$ and every map $g \colon X \to L$ there is exactly one [[lattice homomorphism|BNVRR925X8]] $h \colon F \to L$ with $h \circ \iota = g$.

## 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|6D3WA7N2W4]] 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.