---- > [!definition]+ Definition. ([[reduced word]]) > Let $A$ be a set and $W(A)$ the [[word on a set|words]] on $A$. > > Define the **elementary reduction function** $r:W(A) \to W(A)$ which removes ('cancels') the first occurrence (from left to right) of a pair $a a^{-1}$ or $a ^{-1} a$ from a [[word on a set|word]]. > > We call $w \in W(A)$ a **reduced word** if $r(w)=w$. > > Note that if $w \in W(A)$ has length $n$, then $r^{\lfloor \frac{n}{2} \rfloor}(w)$ is a reduced word. This allows us to define the **reduction map** $\begin{align} R:W(A) \to& W(A) \\ w \mapsto& r^{{\lfloor \frac{\text{length}(w)}{2} \rfloor}}(w) \end{align}$ for turning any word $w$ into its reduced form. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch01