Para-differential Rota-Baxter algebras and their free objects by Gröbner-Shirshov bases
Li Guo, Aniruddha Talele, Shilong Zhang, Shanghua Zheng
TL;DR
This work addresses the algebraic synthesis of FFTC type structures by introducing para-differential Rota-Baxter algebras and providing a complete GS bases framework for free objects. It classifies three operator-type families, develops a bespoke monomial order, proves GS bases for all types, and delivers explicit free algebra constructions, including a noncommutative model and explicit type III realization. The results unify differential and integral operator identities under a cohesive combinatorial method, enabling precise bases for free objects and paving the way for categorical and algebraic applications in differential/integral settings. The approach strengthens the bridge between FFTC inspired algebra and Gröbner-Shirshov theory, with potential impact on combinatorics, operator algebras, and related categorical constructions.
Abstract
The algebraic formulation of the derivation and integration related by the First Fundamental Theorem of Calculus (FFTC) gives rise to the notion of differential Rota-Baxter algebra. The notion has a remarkable list of categorical properties, in terms of the existence of (co)extensions of differential and Rota-Baxter operators, of the lifting of monads and comonads, and of mixed distributive laws. Conversely, using these properties as axioms leads to a class of algebraic structures called para-differential Rota-Baxter algebras. This paper carries out a systematic study of para-differential Rota-Baxter algebras. After their basic properties and examples from Hurwitz series and difference algebras, a Gröbner-Shirshov bases theory is established for para-differential Rota-Baxter algebras. Then an explicit construction of free para-differential Rota-Baxter algebras is obtained.