\begin{definition}[Definition:Reduced Form of Group Word]
Let $X$ be a set.
Let $w$ be a group word on $X$.
The '''reduced form''' $\map {\operatorname {red} } w$ of $w$ is the unique reduced word for which there exists a reduction:
:$w = w^{\paren 0} \to w^{\paren 1} \to \cdots \to w^{\paren n} = \map {\operatorname {red} } w$
\end{definition}