Completeness results for many-valued Lukasiewicz modal systems and relational semantics

The paper is dedicated to the problem of adding a modality to the \Lukasiewicz many-valued logics in the purpose of obtaining completeness results for Kripke semantics. We define a class of modal many-valued logics and their corresponding Kripke models and modal many-valued algebras. Completeness results are considered through the construction of a canonical model. Completeness is obtained for modal finitely-valued logics but also for a modal many-valued system with an infinitary deduction rule. We introduce two classes of frames for the finitely-valued logics and show that they define two distinct classes of Kripke-complete logics.