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:


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

Also known as
Some sources refer to this as the union of $x$ and $y$.

Also see

 * Definition:Meet (Order Theory)
 * Definition:Join Semilattice
 * Definition:Lattice