\begin{definition}[Definition:Join (Order Theory)]
Let $\struct {S, \preceq}$ be an ordered set.
Let $a, b \in S$.
Let their supremum $\sup \set {a, b}$ exist in $S$.
Then the '''join of $a$ and $b$''' is defined as:
:$a \vee b = \sup \set {a, b}$
Expanding the definition of supremum, one sees that $c = a \vee b$ {{iff}}:
:$a \preceq c$ and $b \preceq c$ and $\forall s \in S: a \preceq s \land b \preceq s \implies c \preceq s$
\end{definition}