Fundamental and homogeneous bases of Hopf algebras built from nonsymmetric operads

TL;DR

This paper constructs new bases for Hopf algebras derived from nonsymmetric operads using poset structures on decorated trees, revealing combinatorial properties and basis triples similar to known algebraic structures.

Contribution

It introduces novel partial order structures on free nonsymmetric operads and uses them to define fundamental and homogeneous bases for associated Hopf algebras.

Findings

Posets are EL-shellable with Möbius functions in {-1,0,1}.

Sublattices correspond to m-Fuss-Catalan objects and forests.

Established a basis triple analogous to known Hopf algebra bases.

Abstract

We introduce new partial order structures on the underlying sets of free nonsymmetric operads. These posets involve decorated ordered rooted trees, and their terminal intervals are lattices. These lattices are not graded, not self-dual, and not semi-distributive, but they are EL-shellable, and their M\"bius functions take values in $\{-1, 0, 1\}$. They admit sublattices on the families of $m$-Fuss-Catalan objects and of forests of trees. This latter order structure is used to construct two new bases for the natural Hopf algebras of free nonsymmetric operads: a fundamental basis and a homogeneous basis. Along with the already known elementary basis of these Hopf algebras, this yields a triple of bases. The situation is similar to what is observed in the Hopf algebras of Malvenuto-Reutenauer, Loday-Ronco, and noncommutative symmetric functions, each of which presents such triples of bases and basis changes involving, respectively, the right weak partial order, the Tamari partial order, and the Boolean lattice partial order.