\section{Existence of Subgroup of Big Dipper Semigroup}
Tags: Examples of Subgroups, Big Dipper Operation

\begin{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 $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.
\end{theorem}

\begin{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 {{ProofWiki}}) as the '''big dipper semigroup'''.
Taking the group axioms in turn:
\end{proof}

