---
tag: J2T75ATRV5
kind: concept
name: Join and meet
msc: [06A06, 06B05]
sources:
  - url: https://en.wikipedia.org/wiki/Join_and_meet
same_as:
  wikidata: Q6269005
  wikipedia: Join and meet
---

## Summary

In a [[partially ordered set|0QV2JS9RJ3]], the join of two elements is their least upper bound and the meet is their greatest lower bound.
They are written $x \vee y$ and $x \wedge y$.
Join and meet are the [[supremum|0R3H5PANRQ]] and the [[infimum|0QF1MZ9FRG]] of a two-element set, seen as binary operations.
They are the basic operations of lattice theory: a poset in which all joins and meets of pairs exist is a lattice.

## Explanations

### L1 human

Think of the elements of a poset as points of a diagram, with larger elements drawn higher.
The join of $x$ and $y$ is the lowest point that lies above both; the meet is the highest point below both.
For sets ordered by inclusion, the join of $A$ and $B$ is $A \cup B$ and the meet is $A \cap B$.
For positive integers ordered by divisibility, the join of $4$ and $6$ is $\lcm(4, 6) = 12$ and the meet is $\gcd(4, 6) = 2$.
In some posets a join or a meet can fail to exist.