\begin{definition}[Definition:Big 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 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:
:$m + k n \le a + b$
The operation $+_{m, n}$ is known as the '''Big Dipper'''.
\end{definition}