Congruence Relation on Naturals without Zero for Addition Distinct from Equality is Restricted Dipper Relation

Theorem

Let $\RR^*$ be a congruence relation for addition on the non-zero natural numbers $\N_{>0}$.

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

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

$\RR^* = \RR^*_{m, n}$

where $\RR^*_{m, n}$ denotes the restricted dipper relation.

Proof

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Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.7 \ \text {(c)}$