Congruence Relation on Naturals for Addition Distinct from Equality is Dipper Relation

Theorem

Let $\RR$ be a congruence relation for addition on the natural numbers $\N$.

Let $\RR$ be distinct from the equality relation on $\N$.

Then there exist $m \in \N$ and $n \in \N_{>0}$ such that:

$\RR = \RR_{m, n}$

where $\RR_{m, n}$ denotes the dipper relation.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.6 \ \text {(c)}$