  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.).
