---
tag: WP20S9NZ15
kind: concept
name: Free lattice
msc: [06B25, 08B20]
specializes: [WH4SFAACAS]
sources:
  - url: https://en.wikipedia.org/wiki/Free_lattice
same_as:
  wikidata: Q5500254
  wikipedia: Free lattice
---

## Summary

The free lattice on a set $X$ is the [[lattice|WH4SFAACAS]] 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|BNVRR925X8]] 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|Z86VGP5RMG]].
The free lattice on two generators has four elements, while the free lattice on three generators is already infinite.

## Explanations

### L1 human

Take some letters $x$, $y$, $z$ and form all expressions with $\vee$ and $\wedge$, such as $x \wedge (y \vee z)$.
Two expressions count as equal only when the rules of every lattice force them to be equal.
The result is the free lattice on these letters: the most general lattice that these letters can generate.
With two letters $x$ and $y$ only four elements appear: $x$, $y$, $x \wedge y$ and $x \vee y$.
With three letters, infinitely many different elements appear.