Appendix B - Examples of Coherent Presentations of Monoids

Summary

This appendix presents examples of coherent presentations of monoids. In particular, focus is placed on families of monoids which occur in algebra and whose coherent presentations are computed using the rewriting method that extends Squier’s and Knuth-Bendix’s completion procedures into a homotopical completion-reduction procedure. Coherent presentations of monoids are shown to explicitly describe the actions of monoids on small categories. This construction is applied to the case of Artin monoids. In particular, it is proven that the Zamolodchikov 3-generators extend the Artin presentation into a coherent presentation and, as a byproduct, a constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category is given. Coherent presentations of plactic and Chinese monoids are also provided.