# example of Munn tree

Let $X=\left\{a,b,c\right\}$, $w=aaa^{-1}a^{-1}a^{-1}abb^{-1}ab^{-1}bcaa^{-1}cc^{-1}$. The reduced prefix set of $w$ is

 $\mathrm{red}(\mathrm{pref}(w))=\left\{\varepsilon,a,aa,a^{-1},b,ab^{-1},ac,aca% ,acc\right\}.$

The Munn tree $\mathrm{MT}(w)$ is the following.

 $\xymatrix{&b&{ab^{-1}}\ar[d]_{b}&\\ {a^{-1}}\ar[r]^{a}&\varepsilon\ar[u]^{b}\ar[r]^{a}&a\ar[d]^{c}\ar[r]^{a}&{aa}% \\ &&{ac}\ar[r]^{c}\ar[d]^{a}&{acc}\\ &&{aca}&}$

Note that we have drawn only edges of the form $(v_{1},x,v_{2})$ (i.e. $\xymatrix{v_{1}\ar[r]^{x}&v_{2}}$) with $x\in X$, leaving implicit the existence of the opposite edges $(v_{2},x^{-1},v_{1})$ (i.e. $\xymatrix{v_{2}\ar[r]^{x^{-1}}&v_{1}}$), as usual in the diagram representation of inverse word graphs.

Title example of Munn tree ExampleOfMunnTree 2013-03-22 16:12:02 2013-03-22 16:12:02 Mazzu (14365) Mazzu (14365) 20 Mazzu (14365) Example msc 20M05 msc 20M18