Leibniz rule, locality and supersymmetry on lattice

In a finite volume system, we prove a no-go theorem on a Leibniz rule with a care of locality argument on latttice. The new possibility on the Leibniz rule solutions on lattice is discussed. Although the new solution admits a local difference operator, a non-local product rule is needed. In the case, a supersymmetric interacting theory is simply realized. The difference between finite flavor systems and matrix representations of infinite flavor systems is explained based on a finite volume system analysis including the no-go theorem.