Category:Definitions/Dipper Semigroups

This category contains definitions related to Dipper Semigroups.
Related results can be found in Category:Dipper Semigroups.

Let $m \in \N$ be a natural number.

Let $n \in \N_{>0}$ be a non-zero natural number.

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

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

Definition 1

Let $+_{m, n}$ be the dipper operation on $\N_{< \paren {m \mathop + n} }$:

$\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:

$m + k n \le a + b$

A dipper (semigroup) is a semigroup which is isomorphic to the algebraic structure $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.

Definition 2

Let $\RR_{m, n}$ be the dipper relation on $\N$:

$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}$

Let $\map D {m, n} := \N / \RR_{m, n}$ be the quotient set of $\N$ induced by $\RR_{m, n}$.

Let $\oplus_{m, n}$ be the operation induced on $\map D {m, n}$ by addition on $\N$.

A dipper (semigroup) is a semigroup which is isomorphic to the algebraic structure $\struct {\map D {m, n}, \oplus_{m, n} }$.