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In this paper, we study logics of bounded distributive residuated latticeswith modal operators considering $\Box$ and $\Diamond$ in a noncommutativesetting. We introduce relational semantics for such substructural modal logics.We prove that any canonical logic is Kripke complete via discrete duality andcanonical extensions. That is, we show that a modal extension of thedistributive full Lambek calculus is the logic of its frames if its variety isclosed under canonical extensions. After that, we establish a Priestley-styleduality between residuated distributive modal algebras and topological Kripkestructures based on Priestley spaces.

The Distributive Full Lambek Calculus with Modal Operators