A set-operad of formal fractions and dendriform-like sub-operads

TL;DR

This paper introduces a new operad of formal fractions that encompasses Dendriform and Tridendriform operads, providing a combinatorial description via bi-colored trees and presenting related symmetric operads.

Contribution

It defines a novel set-operad of formal fractions, describes its generators and relations, and offers a combinatorial model using bi-colored trees, extending the understanding of dendriform-like structures.

Findings

Presented a binary generator quadratic relation presentation.

Developed a combinatorial model with bi-colored trees.

Extended results to related symmetric operads.

Abstract

We introduce an operad of formal fractions, abstracted from the Mould operads and containing both the Dendriform and the Tridendriform operads. We consider the smallest set-operad contained in this operad and containing four specific elements of arity two, corresponding to the generators and the associative elements of the Dendriform and Tridendriform operads. We obtain a presentation of this operad (by binary generators and quadratic relations) and an explicit combinatorial description using a new kind of bi-colored trees. Similar results are also presented for related symmetric operads.