Order, lattices, ordered algebraic structures Definition

Join and meet

Builds on Partial order Supremum Infimum

Definition

Let $(P, \le)$ be a partially ordered set and let $x, y \in P$ . The join of $x$ and $y$ is the supremum 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 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 $x \wedge y = x$ .

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Infimum concept
Partial order concept
Supremum concept
Join and meet definition

Tag	`F7YV441H6R`
URI	`m2://F7YV441H6R`
Kind	definition
Writing	W0 stub
MSC	06A06, 06B05
Defines	Join and meet
Uses	Partial order, Supremum, Infimum
Changed	2 November 2026

Join and meet

Definition

Remarks

Reading path

Further reading

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