\begin{definition}[Definition:Reduction of Group Word on Set]
Let $X$ be a set.
Let $w$ be a group word on $X$.
A '''reduction''' of $w$ is a finite sequence of group words $\tuple {w^{\paren 0}, \ldots, w^{\paren n} }$ such that:
:$w^{\paren 0} = w$
:$w_{i + 1}$ is an elementary reduction of $w_i$, for all $i \in \set {0, \ldots, n - 1}$
:$w^{\paren n}$ is reduced
\end{definition}