  We propose a new framework based on the concept of semigroup to understand
the fermion sign problem. By using properties of contraction semigroups, we
obtain new sufficient conditions for quantum lattice fermion models to be
sign-problem-free. Many previous results can be considered as special cases of
our new results. As a direct application of our new results, we construct a new
class of sign-problem-free fermion lattice models, which cannot be understood
by previous frameworks. This framework also provides an interesting aspect in
understanding related quantum many-body systems. We establish a series of
inequalities for all the sign-problem-free fermion lattice models that satisfy
our sufficient conditions.
