  Modal extensions of Product fuzzy logic can be considered both over Kripke
models whose accessibility relation is valued, and over Kripke models with
classical accessibility relation. We study the local consequence of the
previous two modal Product logics. We prove that these logics are standard
complete, in the sense that the logic defined over Kripke models evaluated over
all product algebras coincides with that defined over Kripke models evaluated
over the standard product algebra (with universe [0,1]). This holds both for
the logic over classical Kripke frames, and for that over frames with a valued
accessibility relation. Second, we prove that the previous logics are
decidable.
