Reduced typed angularly decorated planar rooted trees and generalized tridendriform algebras

TL;DR

This paper generalizes tridendriform algebras by introducing a family of products indexed by a set, exploring their structure on Schr{"o}der trees, and connecting them to Rota-Baxter algebras and Koszul duality.

Contribution

It introduces $ Omega$-tridendriform algebras, extends their construction on Schr{"o}der trees, and explores their algebraic and operadic properties.

Findings

Defined $ Omega$-tridendriform algebras with set-indexed products

Established conditions on $ Omega$ for free algebras on Schr{"o}der trees

Connected the new structures to Rota-Baxter algebras and Koszul duality

Abstract

We introduce a generalization of tridendriform algebras, where each of the three products are replaced by a family of products indexed by a set $\Omega$. We study the needed structure on $\Omega$ for free $\Omega$-tridendriform algebras to be built on Schr{\"o}der trees (as it is the case in the classical case), with convenient decorations on their leaves. We obtain in this way extended triassociative semigroups. We describe commutative $\Omega$-tridendriform algebras in terms of typed words. We also study links with generalizations of Rota-Baxter algebras and describe the Koszul duals of the corresponding operads.