# Definition:Restricted Dipper Relation

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## Definition

Let $m, n \in \N_{>0}$ be non-zero natural numbers.

The **restricted dipper relation** $\RR^*_{m, n}$ is the restriction of the **dipper relation** $\RR_{m, n}$ on $\N$:

- $\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}$

## Also see

- Restricted Dipper Relation is Equivalence Relation
- Restricted Dipper Relation is Congruence for Addition
- Restricted Dipper Relation is Congruence for Multiplication

- Results about
**restricted dipper relations**can be found**here**.

## Linguistic Note

The term **restricted dipper relation** was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to be referred to compactly in conjunction with the **dipper semigroup**.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

The term **dipper** was coined by Seth Warner in the context of inductive semigroups.

While Warner takes pains to define this relation with his usual attention to detail, he does not actually assign it a name.

## Sources

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