Definition:Restricted Dipper Relation

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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

  • 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.