Equivalence of Definitions of Dipper Semigroup

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

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

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


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

We have that $\RR_{m, n}$ is an equivalence relation which is compatible with both addition and multiplication.

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 $\N$.

Let $\phi_{m, n}$ be the canonical surjection from $\N$ onto $\map D {m, n}$.

The restriction of $\phi_{m, n}$ to $\N_{< \paren {m + n} }$ is an isomorphism from the semigroup $\struct {\N_{< \paren {m + n} }, +_{m, n} }$ onto $\struct {\map D {m, n}, +_{m, n} }$.

Proof
From Equivalence Relation/Examples/Congruence Modulo Natural Number we have that:
 * $\RR_{m, n}$ is an equivalence relation
 * $\RR_{m, n}$ is compatible with both addition and multiplication.

From Equivalence Relation is Congruence iff Compatible with Operation we have that $\map D {m, n} = \N / \RR_{m, n}$ is properly defined.

By definition of canonical surjection:
 * $\phi_{m, n}: \N_{< \paren {m + n} } \to \map D {m, n}: \map {\phi_{m, n} } x = \eqclass x {\RR_{m, n} }$

By definition $\phi_{m, n}$ is indeed a surjection.

Then we have:

Hence $\phi_{m, n}$ is an injection.

Thus $\phi_{m, n}$ is by definition a bijection

It remains to show that $\phi_{m, n}$ has the morphism property.