Definition:Dipper Operation

Definition

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

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

Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:

$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$

The dipper operation $+_{m, n}$ is the binary operation on $\N_{< \paren {m \mathop + n} }$ defined as:

$\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b = \begin{cases}

a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying:

$m + k n \le a + b$

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 $a +_{m, n} b$ can be interpreted as:

Start at $\text{Alkaid}$ and count $a$ stars along the handle and then clockwise round the pan.

Then count $b$ stars more from there.

$a +_{m, n} b$ is the number of the star you land on.

Examples

Example: $m = 0$

Let $m = 0$.

Then $+_{m, n}$ degenerates to modulo addition modulo $n$ on $\N_{<n}$:

$\forall a, b \in \N_{<n}: a +_n b = a + b - k n$

where $k$ is the largest integer satisfying:

$k n \le a + b$

Example: $n = 1$

Let $n = 1$.

Then $+_{m, n}$ degenerates to the following operation on $\N_{< \paren {m \mathop + n} }$:

$\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, 1} b = \begin{cases}

a + b & : a + b < m \\ m & : a + b \ge m \end{cases}$

Example: $+_{3, 4}$

Let $m = 3$ and $n = 4$.

The Cayley table for $+_{3, 4}$ can be presented as follows:

$\begin{array}{r|rrrrrrr}

+_m & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 3 \\ 2 & 2 & 3 & 4 & 5 & 6 & 3 & 4 \\ 3 & 3 & 4 & 5 & 6 & 3 & 4 & 5 \\ 4 & 4 & 5 & 6 & 3 & 4 & 5 & 6 \\ 5 & 5 & 6 & 3 & 4 & 5 & 6 & 3 \\ 6 & 6 & 3 & 4 & 5 & 6 & 3 & 4 \\ \end{array}$

Also see

• Dipper Operation is Associative
• Dipper Operation is Commutative

• Results about dipper operations can be found here.

Linguistic Note

The term dipper operation 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.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.8$