Binary Operation on Natural Numbers on which Congruence Relations induce Convex Equivalence Classes

Theorem
Let $\N$ denote the set of natural numbers: $\set {0, 1, 2, \ldots}$

Let $\circ$ be a binary operation on $\N$ with the following properties:


 * $\paren {\text O 1}: \quad$ $\circ$ has an identity element $e$
 * $\paren {\text O 2}: \quad$ Every equivalence relation $\RR$ on $\N$ whose equivalence classes are convex subsets of $\N$ is a congruence relation for $\circ$.

Then $\circ$ is the max operation on $\N$:
 * $\forall a, b \in \N: a \circ b = \max \set {a, b}$