Coherent presentations of a class of monoids admitting a Garside family

Pierre-Louis Curien (1) , Alen Ðurić (1) , Yves Guiraud (2, 3)

1 PI.R2 - Design, study and implementation of languages for proofs and programs
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs

This paper shows how to construct coherent presentations of a class of monoids, including left-cancellative noetherian monoids containing no nontrivial invertible element and admitting a Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of Deligne's original construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to a dual braid monoid, and to some monoids which are neither Artin-Tits nor Garside. For the Artin-Tits monoid of type $\widetilde{A}_{2}$, a finite coherent presentation is given, having a finite Garside family as a generating set.

Mots clés: Monoid; Coherent presentation; Higher rewriting; Polygraph; Artin-Tits monoid; Garside family

Domaines: Théorie des groupes [math.GR]; Catégories et ensembles [math.CT]

Pierre-Louis Curien, Alen Ðurić, Yves Guiraud. Coherent presentations of a class of monoids admitting a Garside family. 2021. ⟨hal-03276119v1⟩