Definition:Reduction of Group Word on Set

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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

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