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 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 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 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:
 * $m + k n \le a + b$

Taking the group axioms in turn:

Let $a, b \in H$.

Then:
 * $2 m \le a + b \le 2 \paren {m + n - 1}$

Thus in all cases $a + b > m$.

Thus by definition of $+_{m, n}$:
 * $a +_{m, n} b = a + b - k n$

where $k$ is the largest such that $m + k n < a + b$

Hence:

That is:
 * $a +_{m, n} b \in H$

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:


 * $m \le a < m + n$

Hence
 * $a +_{m, n} n = a$

From Dipper Semigroup is Commutative Semigroup we have that:


 * $n +_{m, n} a = a +_{m, n} n = n$

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

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

Hence we need to find $x \in H$ such that:
 * $a +_{m, n} x = n$

That is:


 * $a + x - k n = n$

So:
 * $x = \paren {k + 1} n - a$

such that $m \le x < m + n$

where:
 * $m - n + a \le k n < m + a$

Thus every element $x$ of $\struct {H, +_{m, n} }$ has an inverse $x^{-1}$ where:
 * $\paren {k + 1} n - a$

such that:
 * $m - n + a \le k n < m + a$

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