Tridendriform algebras on hypergraph polytopes

TL;DR

This paper generalizes the tridendriform algebra structure from associahedra and permutohedra to a broader class of hypergraph polytopes, introducing a new polydendriform algebra that extends the shuffle product.

Contribution

It introduces the polydendriform algebra, extending the tridendriform structure to hypergraph polytopes and generalizing the shuffle product to multiple arguments.

Findings

Extended tridendriform decomposition to new hypergraph polytopes.

Defined the polydendriform algebra structure.

Unified algebraic structures for various polytopes.

Abstract

We extend the works of Loday-Ronco and Burgunder-Ronco on the tridendriform decomposition of the shuffle product on the faces of associahedra and permutohedra, to other families of hypergraph polytopes (or nestohedra), including simplices, hypercubes and some new families. We also extend the shuffle product to take more than two arguments, and define accordingly a new algebraic structure, that we call polydendriform, from which the original tridendriform equations can be crisply synthesized.