Free $\Omega$-Rota-Baxter systems and Gr\"obner-Shirshov bases

TL;DR

This paper introduces the concept of $\Omega$-Rota-Baxter systems, generalizing existing algebraic structures, and constructs their free objects using Gr"obner-Shirshov bases within operated algebras.

Contribution

It defines $\Omega$-Rota-Baxter systems, develops a basis for free such systems, and unifies various Rota-Baxter related structures through a new construction method.

Findings

Established a linear basis for free $\Omega$-Rota-Baxter systems

Unified construction of free Rota-Baxter related algebras

Introduced new methods using Gr"obner-Shirshov bases

Abstract

In this paper, we propose the concept of an $\Omega$-Rota-Baxter system, which is a generalization of a Rota-Baxter system and an $\Omega$-Rota-Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis of a free $\Omega$-Rota-Baxter system for an extended diassociative semigroup $\Omega$, in terms of bracketed words and the method of Gr\"obner-Shirshov bases. As applications, we introduce the concepts of Rota-Baxter system family algebras and matching Rota-Baxter systems as special cases of $\Omega$-Rota-Baxter systems, and construct their free objects. Meanwhile, free $\Omega$-Rota-Baxter algebras of weight zero, free Rota-Baxter systems, free Rota-Baxter family algebras and free matching Rota-Baxter algebras are reconstructed via new method.