# Definition:Join (Order Theory)

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b \in S$.

Let their supremum $\sup \left\{{a, b}\right\}$ exist in $S$.

Then the join of $a$ and $b$ is defined as:

$a \vee b = \sup \left\{{a, b}\right\}$

Expanding the definition of supremum, one sees that $c = a \vee b$ if and only if:

$a \preceq c$ and $b \preceq c$ and $\forall s \in S: a \preceq s \land b \preceq s \implies c \preceq s$

## Also known as

Some sources refer to this as the union of $a$ and $b$.

## Also see

• Results about join (and meet) can be found here.