Definition:Dipper Semigroup/Definition 2

Definition
Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.

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 $+_{m, n}$ 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 $\struct {\map D {m, n}, +_{m, n} }$.

Also see

 * Equivalence of Definitions of Dipper Semigroup