---
tag: WH4SFAACAS
kind: concept
name: Lattice
msc: [06B05]
specializes: [74X7JRG2J7]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(order)
same_as:
  wikidata: Q595364
  wikipedia: Lattice (order)
---

## Summary

A lattice is a [[partially ordered set|0QV2JS9RJ3]] in which every two elements have a [[join and a meet|J2T75ATRV5]].
These are a least upper bound and a greatest lower bound.
Equivalently, it is a set with two binary operations $\vee$ and $\wedge$ that are commutative and associative and satisfy the absorption laws.
Lattices occur all over mathematics.
Subsets of a set, subgroups of a group, subspaces of a vector space and open sets of a space all form lattices.
Special classes such as [[distributive lattices|0HM0HQCK6N]] and [[Boolean algebras|0XADJKM699]] are central in algebra and logic.

## Explanations

### L1 human

A lattice is an ordering in which any two elements have a best common upper bound and a best common lower bound.
For the subsets of a set ordered by inclusion, these are the union and the intersection.
For positive integers ordered by divisibility, they are the least common multiple and the greatest common divisor.
In a picture of the order, from any two points one can go up to a lowest common point and down to a highest common point.
Not every ordering is a lattice: two incomparable elements with two incomparable common upper bounds have no best one.