Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis

Joan Bellier-Millès (1) , Bérénice Delcroix-Oger (2) , Eric Hoffbeck (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
3 LAGA - Laboratoire Analyse, Géométrie et Applications

In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincaré-Birkhoff-Witt basis. Furthermore, we give counterexamples to the converse.

Mots clés: Operad; Partition poset; Rewriting

Domaines: Topologie algébrique [math.AT]

Joan Bellier-Millès, Bérénice Delcroix-Oger, Eric Hoffbeck. Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis. 2018. ⟨hal-01937173v1⟩