Equivalence of Definitions of Dipper Semigroup/Examples

Examples of Use of Equivalence of Definitions of Dipper Semigroup

Example: $m = 0$

Let $+_n$ be the operation on $\N_{<n$ defined as:

$\forall a, b \in \N_{<n}: a +_n b = a + b - k n$

where $k$ is the largest integer satisfying:

$k n \le a + b$

Let $\RR_n$ be the relation on $\N$ defined as:

$\forall x, y \in \N: x \mathrel {\RR_n} y \iff n \divides \size {x - y}$

Let $\map D n := \N / \RR_n$ be the quotient set of $\N$ induced by $\RR_n$.

Let $\oplus_n$ be the operation induced on $\map D n$ by addition on $\N$.

Then $\struct {\N_{<n}, +_n}$ is isomorphic to $\struct {\map D n, \oplus_n}$.