Equivalence of Definitions of Dipper Semigroup/Examples
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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 $\struct {\map D n, \oplus_n}$.