Join Semilattice is Ordered Structure

Theorem
Let $\left({S, \vee, \preceq}\right)$ be a join semilattice.

Then $\left({S, \vee, \preceq}\right)$ is an ordered structure.

That is, $\preceq$ is compatible with $\vee$.

Also see

 * Meet Semilattice is Ordered Structure