Operated semigroups, Motzkin paths and rooted trees

TL;DR

This paper develops an algebraic framework for combinatorial structures like rooted trees and Motzkin paths, providing new insights and constructions for free Rota-Baxter algebras.

Contribution

It introduces operated semigroups as a unifying algebraic framework for combinatorial objects, enabling algebraic interpretation and construction of Rota-Baxter algebras.

Findings

Operated semigroups offer a combinatorial-algebraic bridge.

Constructs free Rota-Baxter algebras using Motzkin paths and rooted trees.

Provides algebraic descriptions of recursive combinatorial structures.

Abstract

Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic framework of operated semigroups. This framework provides the concept of operated semigroups with intuitive and convenient combinatorial descriptions, and at the same time endows the familiar combinatorial objects with a precise algebraic interpretation. As an application, we obtain constructions of free Rota-Baxter algebras in terms of Motzkin paths and rooted trees.