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
- Results about restricted dipper operations can be found here.
Linguistic Note
The term restricted 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.