---
tag: 0T36P6TXQ6
kind: definition
name: Distributive lattice
msc: [06D05]
defines: 0HM0HQCK6N
sources:
  - url: https://en.wikipedia.org/wiki/Distributive_lattice
    locator: Definition
---

## Definition

A lattice $(L, \vee, \wedge)$ is distributive if $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$ for all $x, y, z \in L$.

## Remarks

The condition is equivalent to its dual: $x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$ for all $x, y, z$.
Every totally ordered set is a distributive lattice, with $\max$ as join and $\min$ as meet. Every [[Boolean algebra|06GKBTAA5E]] is a distributive lattice.
The diamond lattice $M_3$ and the pentagon lattice $N_5$ are the basic lattices that are not distributive.

## Explanations

### L1 human

A lattice is distributive when meet distributes over join, like intersection over union for sets.
Then join also distributes over meet.