Condition for Equivalence Relation for Max Operation on Natural Numbers to be Congruence

Theorem
Let $\RR$ be an equivalence relation on the set of natural numbers $\NN$.

Let $\vee$ denote the max operation on $\N$:
 * $\forall a, b \in \N: a \vee b := \max \set {a, b}$

Then:
 * $\RR$ is a congruence relation for $\vee$ on $\N$


 * every equivalence class under $\RR$ is a convex subset of $\N$.
 * every equivalence class under $\RR$ is a convex subset of $\N$.