Coherent presentation for the hypoplactic monoid of rank n

In this thesis, we construct a coherent presentation for the hypoplactic monoid of rank n and characterize the confluence diagrams associated with it, then we use the theory of quasi-Kashiwara operators and quasi-crystal graphs to prove that all confluence diagrams can be obtained from those diagrams whose vertices are highest-weight words. To do so, we first give a complete rewriting system for the hypoplactic monoid of rank n, then, using an extension of the Knuth–Bendix completion procedure called the homotopical completion procedure, we compute the previously mentioned coherent presentation, which, from a viewpoint of Monoidal Category Theory, gives us a family of generators of the relations amongst the relations. These coherent presentations are used for representations of monoids and are particularly useful to describe actions of monoids on categories. The theoretical background is given without proof, since the main purpose of this thesis is to present new results