Homothetic Rota-Baxter systems and Dyck$^m$-algebras

TL;DR

This paper explores the relationship between generalized Rota-Baxter operators, Rota-Baxter systems enriched with homothetisms, and their connection to Dyck$^m$-algebras, expanding the algebraic framework in this area.

Contribution

It demonstrates that generalized Rota-Baxter operators are a special case of Rota-Baxter systems with homothetisms, which can produce Dyck$^m$-algebras, providing new algebraic insights.

Findings

Generalized Rota-Baxter operators are a special case of Rota-Baxter systems.

Enrichment of Rota-Baxter systems with homothetisms leads to Dyck$^m$-algebras.

The work broadens the understanding of algebraic structures related to Rota-Baxter operators.

Abstract

It is shown that generalized Rota-Baxter operators introduced in [W.A. Martinez, E.G. Reyes & M. Ronco, Int. J. Geom. Meth. Mod. Phys. 18 (2021) 2150176] are a special case of Rota-Baxter systems [T. Brzezi\'nski, J. Algebra 460 (2016), 1-25]. The latter are enriched by homothetisms and then shown to give examples of Dyck$^m$-algebras.