\begin{definition}[Definition:Reduced Group Word on Set]
Let $S$ be a set.
Let $n \ge 0$ be a natural number.
Let $w = w_1 \cdots w_i \cdots w_n$ be a group word on $S$ of length $n$.
Then $w$ is '''reduced''' {{iff}}:
:$\forall i \in \set {1, 2, \ldots, n - 1}: w_i \ne {w_{i + 1} }^{-1}$
\end{definition}