---
tag: F7YV441H6R
kind: definition
name: Join and meet
msc: [06A06, 06B05]
defines: J2T75ATRV5
uses: [0QV2JS9RJ3, 0R3H5PANRQ, 0QF1MZ9FRG]
sources:
  - url: https://en.wikipedia.org/wiki/Join_and_meet
---

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $x, y \in P$.
The *join* of $x$ and $y$ is the [[supremum|0R3H5PANRQ]] of $\set{x, y}$ in $P$, when it exists.
So it is an element $j \in P$ with $x \le j$ and $y \le j$, such that $j \le z$ for every $z \in P$ with $x \le z$ and $y \le z$.
It is written $x \vee y$.
The *meet* of $x$ and $y$ is the [[infimum|0QF1MZ9FRG]] of $\set{x, y}$ in $P$, when it exists.
It is written $x \wedge y$.
More generally, the join $\bigvee S$ and the meet $\bigwedge S$ of a subset $S \subseteq P$ are its supremum and its infimum.

## Remarks

Example: in the power set of a set $X$, ordered by $\subseteq$, the join of $A$ and $B$ is $A \cup B$.
Indeed $A \cup B$ contains $A$ and $B$, and every set that contains $A$ and $B$ contains $A \cup B$.
Dually, the meet is $A \cap B$.
Non-example: in the poset $\set{a, b, c, d}$ with $a < c$, $a < d$, $b < c$, $b < d$ and no other strict relations, the elements $a$ and $b$ have no join.
They have two upper bounds, $c$ and $d$, and neither is below the other, so there is no least upper bound.
When they exist, joins and meets are unique, and $x \le y$ holds exactly when $x \vee y = y$, and exactly when $x \wedge y = x$.