\begin{definition}[Definition:Join Semilattice/Definition 2]
Let $\struct {S, \vee}$ be a semilattice.
Let $\preceq$ be the ordering on $S$ defined by:
:$a \preceq b \iff \paren {a \vee b} = b$
Then the ordered structure $\struct {S, \vee, \preceq}$ is called a '''join semilattice'''.
\end{definition}