Construction of free commutative integro-differential algebras by the method of Gr\"{o}bner-Shirshov bases

TL;DR

This paper develops a method to construct canonical bases for free commutative integro-differential algebras using Gr"obner-Shirshov bases, extending algebraic tools to these structures.

Contribution

It introduces a systematic approach to build bases for free commutative integro-differential algebras via Gr"obner-Shirshov bases and generalizes functional derivations.

Findings

Established the Composition-Diamond Lemma for these algebras.

Obtained a weakly monomial order enabling basis construction.

Generalized functional derivations for arbitrary weights.

Abstract

In this paper, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gr\"obner-Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota-Baxter algebras of order $n$. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gr\"{o}bner-Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional derivations to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.