On the local consequence of modal Product logic: standard completeness and decidability

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.