Algebraic and combinatorial structures on pairs of twin binary trees

TL;DR

This paper introduces a new Hopf algebra structure on Baxter combinatorial objects, using a Baxter monoid and a Robinson-Schensted-like correspondence, revealing algebraic properties like freeness and self-duality.

Contribution

It constructs a novel Hopf algebra based on Baxter objects, extending algebraic frameworks with new monoid and correspondence definitions.

Findings

Defined the Baxter monoid and lattice structure on pairs of twin binary trees.

Proved the Hopf algebra is free and self-dual.

Provided multiplicative bases for the algebra.

Abstract

We give a new construction of a Hopf algebra defined first by Reading whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e., Baxter permutations, pairs of twin binary trees, etc.). Our construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like correspondence and insertion algorithm. Indeed, the Baxter monoid leads to the definition of a lattice structure over pairs of twin binary trees and the definition of a Hopf algebra. The algebraic properties of this Hopf algebra are studied and among other, multiplicative bases are provided, and freeness and self-duality proved.