\begin{definition}[Definition:Join Semilattice/Definition 1]
Let $\struct {S, \preceq}$ be an ordered set.
Suppose that for all $a, b \in S$:
:$a \vee b \in S$
where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.
Then the ordered structure $\struct {S, \vee, \preceq}$ is called a '''join semilattice'''.
\end{definition}