Dualities for Some Intuitionistic Modal Logics

We present a duality for the intuitionistic modal logic IK introduced by Fischer Servi in [8, 9]. Unlike other dualities for IK reported in the literature (see for example [13]), the dual structures of the duality presented here are ordered topological spaces endowed with just one extra relation, which is used to define the set-theoretic representation of both 2 and 3. Also, this duality naturally extends the definitions and techniques used by Fischer Servi in the proof of completeness for IK via canonical model construction [10]. We also give a parallel presentation of dualities for the intuitionistic modal logics IntK2 and IntK3. Finally, we turn to the intuitionistic modal logic MIPC, which is an axiomatic extension of IK, and we give a very natural characterization of the dual spaces for MIPC introduced in [2] as a subcategory of the category of the dual spaces for