---
tag: T5GK6DH8BR
kind: concept
name: Free distributive lattice
msc: ["06B25", "06D05"]
specializes: [0HM0HQCK6N]
sources:
  - url: https://en.wikipedia.org/wiki/Free_lattice
---

## Summary

The free distributive lattice on a set $S$ is the [[distributive lattice|0HM0HQCK6N]] generated by $S$ with no extra relations.
The only equalities in it are those that the lattice laws and distributivity force.
Every map from $S$ into a distributive lattice extends to exactly one lattice homomorphism from it.
Unlike the [[free lattice|WP20S9NZ15]] on three or more generators, the free distributive lattice on a finite set is finite.
On $n$ generators it has $M(n) - 2$ elements, where $M(n)$ is the [[Dedekind number|481ZPQ0GC3]].

## Explanations

### L1 human

Start with letters $x, y, z, \ldots$ and build all expressions with "and" and "or".
Two expressions count as equal when the rules of a distributive lattice prove them equal, as in ordinary logic without negation and without constants.
Every expression can then be written as an "or" of "and"s of letters.
With two letters there are exactly four different elements: $x \wedge y$, $x$, $y$ and $x \vee y$.