Definition:Restricted Dipper Operation

Definition
Let $m, n \in \N_{>0}$ be non-zero natural numbers.

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 $\N^*_{< \paren {m \mathop + n} }$ denote the set defined as $\N_{< \paren {m \mathop + n} } \setminus \set 0$:
 * $\N^*_{< \paren {m \mathop + n} } := \set {1, 2, \ldots, m + n - 1}$

The restricted 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:
 * $m + k n \le a + b$

Also see

 * Restricted Dipper Operation is Associative
 * Restricted Dipper Operation is Commutative