Definition:Elementary Reduction of Group Word on Set

Definition
Let $X$ be a set.

Let $X^\pm$ be a set of literals on $X$.

Let $v$ and $w$ be formal words on $X^\pm$.

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

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

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

Also see

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