Definition:Elementary Reduction of Group Word on Set

Definition
Let $X$ be a set.

Let $v$ and $w$ be group words on $X$.

Let $n$ be the length of $v$.

Then $w$ is an elementary reduction of $v$ :
 * $w$ has length $n - 2$
 * There exists $k \in \set {1, \ldots, n - 1}$ such that:
 * $v_k = v_{k + 1}^{-1}$
 * $w_i = \begin {cases} v_i & : i < k \\ v_{i + 2} & : i > k + 1 \end {cases}$

This is denoted $v \overset 1 \longrightarrow w$.

Also see

 * Definition:Reduced Group Word on Set
 * Definition:Reduction of Group Word on Set