Generalizing Dendriform Algebras: Dyck(M)-Algebras, Rota(M)-Algebras, And Rota-Baxter Operators

We review the notion of a Dyckm-algebra, an algebraic structure introduced recently by López, Préville-Ratelle and Ronco during their work on the splitting of associativity via m-Dyck paths, and we also introduce Rotam-algebras: both structures can be considered as generalizations of dendriform structures. We obtain examples of Dyckm-algebras in terms of planar rooted binary trees equipped with a particular type of Rota-Baxter operator, and we present examples of Rotam-algebras using left averaging morphisms. As an application, we observe that the structures presented here allow us to introduce quite naturally a “non-associative version”of the Kadomtsev-Petviashvili hierarchy.