Dipper Operation/Examples/m = 0
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Examples of Dipper Operation
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:
- $m + k n \le a + b$
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:
- $k n \le a + b$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.8 \ \text{(a)}$