Gr\"obner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras (unabridged version)

TL;DR

This paper develops Gr"obner-Shirshov bases for free quadri- and dendriform algebras using operadic methods and planar rooted trees, simplifying previous complex computations.

Contribution

It introduces an intuitive operadic approach to compute Gr"obner-Shirshov bases for non-symmetric operads of dendriform and quadri-algebras, improving upon prior methods.

Findings

Successfully computed Gr"obner-Shirshov bases for free quadri-algebras.

Simplified the derivation of bases for dendriform algebras compared to previous work.

Demonstrated the effectiveness of planar rooted trees in operadic computations.

Abstract

In this paper we use the operadic framework to find Gr\"obner-Shirshov bases for the free quadri-algebra. We perform computations using the representation of the nonsymmetric operad by planar rooted trees in a very intuitive way. Gr\"obner-Shirshov bases for the free dendriform algebra are also found with this technique, simplifying the work by Chen and Wang in 2010.