  A generalized criterion for signature related algorithms to compute Gr\"obner
basis is proposed in this paper. Signature related algorithms are a popular
kind of algorithms for computing Gr\"obner basis, including the famous F5
algorithm, the extended F5 algorithm and the GVW algorithm. The main purpose of
current paper is to study in theory what kind of criteria is correct in
signature related algorithms and provide a generalized method to develop new
criteria. For this purpose, a generalized criterion is proposed. The
generalized criterion only relies on a general 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 related algorithms. For {\em admissible} partial orders,
a complete proof of the correctness of the algorithm based on this generalized
criterion is also presented. This proof has no extra requirements on the
computing order of critical pairs, and is also valid for non-homogeneous
polynomial systems. More importantly, the partial orders implied by existing
criteria are admissible. Besides, one can also check whether a new criterion is
correct in signature related algorithms or even develop new criteria by using
other admissible partial orders in the generalized criterion.
