Semantic Analysis of Subexponential Modalities in Distributive Non-commutative Linear Logic

In this paper, we consider the full Lambek calculus enriched with subexponential modalities in a distributive setting. We show that the distributive Lambek calculus with subexponentials is complete with respect to its Kripke frames via canonical extensions. In this approach, we consider subexponentials as S4-like modalities and each modality is interpreted with a reflexive and transitive relation similarly to usual Kripke semantics.