  A generalized criterion for signature-based algorithms to compute Gr\"obner
bases is proposed in this paper. This criterion is named by "generalized
criterion", because it can be specialized to almost all existing criteria for
signature-based algorithms which include the famous F5 algorithm, F5C, extended
F5, G$^2$V and the GVW algorithm. The main purpose of current paper is to study
in theory which kind of criteria is correct in signature-based algorithms and
provide a generalized method to develop new criteria. For this purpose, by
studying some key facts and observations of signature-based algorithms, a
generalized criterion is proposed. The generalized criterion only relies on a
partial order defined on a set of polynomials. When specializing the partial
order to appropriate specific orders, the generalized criterion can specialize
to almost all existing criteria of signature-based algorithms. For {\em
admissible} partial orders, a proof is presented for the correctness of the
algorithm that is based on this generalized criterion. And the partial orders
implied by the criteria of F5 and GVW are also shown to be admissible. More
importantly, the generalized criterion provides an effective method to check
whether a new criterion is correct as well as to develop new criteria for
signature-based algorithms.
