Dipper Operation/Examples

Examples of Dipper Operations

Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.

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

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

Let $+_{m, n}$ be the dipper 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:

$m + k n \le a + b$

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}$