Generalized dendrifom algebras and typed binary trees

We here both unify and generalize nonassociative structures on typed binary trees, that is to say plane binary trees which edges are decorated by elements of a set $Ω$. We prove that we obtain such a structure, called an $Ω$-dendriform structure, if $Ω$ has four products satisfying certain axioms (EDS axioms), including the axioms of a diassociative semigroup. This includes matching dendriform algebras introduced by Zhang, Gao and Guo and family dendriform algebras associated to a semigroup introduced by Zhang, Gao and Manchon , and of course dendriform algebras when $Ω$ is reduced to a single element. We also give examples of EDS, including all the EDS of cardinality two; a combinatorial description of the products of such a structure on typed binary trees, but also on words; a study of the Koszul dual of the associated operads; and considerations on the existence of a coproduct, in order to obtain dendriform bialgebras.

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