# Definition:Dipper Relation

## Definition

Let $m \in \N$ be a natural number.

Let $n \in \N_{>0}$ be a non-zero natural number.

The dipper relation $\RR_{m, n}$ is the relation on $\N$ defined as:

$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}$

### Illustration When the stars of the Big Dipper are numbered as shown, the sequence:

$1, 1 +_{3, 4} 1, 1 +_{3, 4} 1 +_{3, 4} 1, \ldots$

traces out those stars in the order:

first the handle: $\text{Alkaid}, \text{Mizar}, \text{Alioth}$

then:

round the pan indefinitely: $\text{Megrez}, \text{Dubhe}, \text{Merak}, \text{Phecda}, \text{Megrez}, \ldots$

Hence $x \mathrel {\RR_{m, n} } y$ can be interpreted as:

Start at $\text{Alkaid}$ and count $x$ stars along the handle and then clockwise round the pan.
Then start at $\text{Alkaid}$ again and count $y$ stars along the handle and then clockwise round the pan.
$x \mathrel {\RR_{m, n} } y$ if and only if you end up at the same star.

## Also see

• Results about dipper relations can be found here.

## Linguistic Note

The term 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.