Typed binary trees and generalized dendrifom algebras (Q2049365)

Dendriform algebras are associative algebras with an associativity splitting, that is to say their associative product can be written as a sum of two products subject to some axioms. Classical examples of dendriform algebras are given by shuffle algebras. Other examples include the dendriform algebras based on binary tries, on \(\Omega\)-typed binary tries (i.e., on trees with a map from this set of edges to some set \(\Omega\)). The aim in the article is to give both a unification and a generalization of the extended dendriform structures. A notion of an \(\Omega\)-dendriform algebra is introduced. The constructed objects are studied also from a Hopf-algebraic and a combinatorial point of view.