Axioms for a theory of signature bases

TL;DR

This paper introduces an axiomatic framework for signature-based Gr"obner bases, aiming to clarify their theory and facilitate adaptation to various algebraic settings beyond the classical commutative case.

Contribution

It proposes a general axiomatic approach that separates the theoretical foundations from specific algorithms, enabling broader applicability and understanding.

Findings

Provides a unified axiomatic framework for signature-based Gr"obner bases

Enables adaptation to noncommutative and non-Noetherian settings

Facilitates the development of new algorithms based on the axioms

Abstract

Twenty years after the discovery of the F5 algorithm, Gr\"obner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gr\"obner bases with signatures with the purpose of uncoupling the theory and the algorithms, and giving general results applicable in many different settings (e.g. Gr\"obner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.).