Editing Knuth–Bendix completion algorithm (section)

==== A non-terminating example ====
The order of the generators may crucially affect whether the Knuth–Bendix completion terminates. As an example, consider the [[free Abelian group]] by the monoid presentation:

: <math>\langle x, y, x^{-1}, y^{-1}\, |\, xy=yx, xx^{-1} = x^{-1}x = yy^{-1} = y^{-1}y = 1 \rangle  .</math>

The Knuth–Bendix completion with respect to lexicographic order <math>x < x^{-1} < y < y^ {-1}</math> finishes with a convergent system, however considering the length-lexicographic order <math>x < y < x^{-1} < y^{-1}</math> it does not finish for there are no finite convergent systems compatible with this latter order.<ref name="BogopolskiBumagin2011">{{cite book |editor=Oleg Bogopolski |editor2=Inna Bumagin |editor3=Olga Kharlampovich |editor4=Enric Ventura|title=Combinatorial and Geometric Group Theory: Dortmund and Ottawa-Montreal conferences|year=2011|publisher=Springer Science & Business Media|isbn=978-3-7643-9911-5|page=62|chapter=Geodesic Rewriting Systems and Pregroups|author=V. Diekert |author2=A.J. Duncan |author3=A.G. Myasnikov}}</ref>