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.).

翻译：在F5算法发现二十年后的今天，带签名的Gröbner基依然难以理解且难以适应不同场景。这与Buchberger算法形成鲜明对比——后者可在保持正确性与终止性显见的前提下进行多方向调整。本文提出一种带签名Gröbner基的公理化方法，旨在将理论框架与算法实现解耦，并给出能适用于多种不同场景（例如子模的Gröbner基、F4型约化、非交换环、非Noetherian环境等）的通用结论。