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

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## 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 $\N$:

- $\forall a, b \in \N: a \circ b = \max \set {a, b}$

## Proof

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## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.20 \ \text {(b)}$