Para-differential Rota-Baxter algebras and their free objects by Gr\"obner-Shirshov bases

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 Gr\"obner-Shirshov bases theory is established for para-differential Rota-Baxter algebras. Then an explicit construction of free para-differential Rota-Baxter algebras is obtained.