Cover Systems for the Modalities of Linear Logic

Ono’s modal FL-algebras are models of an extension of Full Lambek logic that has the modalities ! and ? of linear logic. Here we define a notion of modal FL-cover system that combines aspects of Beth-Kripke-Joyal semantics with Girard’s interpretation of the ! modality, and has structured subsets that interpret propositions. We show that any modal FL-algebra can be represented as an algebra of propositions of some modal FL-cover system.