# Free Rota-Baxter systems and a Hopf algebra structure

## TL;DR

This paper constructs a linear basis for free Rota-Baxter systems using Gröbner-Shirshov bases and establishes a left counital Hopf algebra structure on them, advancing algebraic understanding.

## Contribution

It introduces a new linear basis for free Rota-Baxter systems and defines a Hopf algebra structure on them, combining combinatorial and algebraic techniques.

## Key findings

- Established a linear basis for free Rota-Baxter systems
- Defined a left counital Hopf algebra structure on these systems
- Applied Gröbner-Shirshov bases method to algebraic structures

## Abstract

In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gr\"{o}bner-Shirshov bases method and then we obtain a left counital Hopf algebra structure on a free Rota-Baxter system.