Existence of Subgroup of Dipper Semigroup

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

Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the big dipper semigroup.

Consider the subset $H \subseteq N_{< \paren {m \mathop + n} }$ defined as:
 * $H = \set {k \in \N: m \le k < m + n} = \set {m, m + 1, \ldots, m + n - 1}$

Then the substructure $\struct {H, +_{m, n} }$ is a subgroup of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.

Proof
Recall the definition of the big dipper semigroup:

Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
 * $\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$

Let $+_{m, n}$ be the big dipper operation on $\N_{< \paren {m \mathop + n} }$:
 * $\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b = \begin{cases}

a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying:
 * $m + k n \le a + b$

The algebraic structure $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ is known (on ) as the big dipper semigroup.

Taking the group axioms in turn:

Let $x, y \in H$.

Then:
 * $2 m \le x + y \le 2 \paren {m + n - 1}$

Thus in all cases $x + y < m$.

Then:
 * $m \le x + y - k n < m + n - 1$

and so $x +_{m, n} y \in H$.

Thus $x +_{m, n} y \in H$ and so $\struct {H, +_{m, n} }$ is closed.

From Restriction of Associative Operation is Associative, associativity is inherited by $\struct {H, +_{m, n} }$ from $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.

We have:

Thus $n$ is the identity element of $\struct {H, +_{m, n} }$.

We have that $n$ is the identity element of $\struct {H, +_{m, n} }$.

Thus every element $x$ of $\struct {H, +_{m, n} }$ has an inverse $...$.

All the group axioms are thus seen to be fulfilled, and so $\struct {H, +_{m, n} }$ is a group.