Definition:Restricted Dipper Semigroup

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

Let $\RR^*_{m, n}$ be the restricted dipper relation on $\N$:


 * $\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}$

Let $\map {D^*} {m, n} := \N_{>0} / \RR^*_{m, n}$ be the quotient set of $\N_{>0}$ induced by $\RR^*_{m, n}$.

Let $\oplus^*_{m, n}$ be the operation induced on $\map {D^*} {m, n}$ by addition on $\N_{>0}$.

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