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 $v$ if and only if:

$w$ 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 $v \overset 1 \longrightarrow w$.