Structure theorems for dendriform and tridendriform algebras

Abstract

We state new Cartier–Milnor–Moore Poincaré–Birkhoff–Witt theorems for dendriform and tridendriform structures. We introduce the terplicial coalgebra structure as an analogue in the tridendriform algebras of the duplicial co-structure for the dendriform case, and prove a rigidity theorem.