Generality of Humberstone frames versus Kripke and full possibility frames

Determine whether the class of Humberstone possibility frames—defined by P being the set of regular open sets and satisfying the frame conditions up, R, and R—characterizes exactly the same normal modal logics as the class of full possibility frames, and whether it yields a strictly larger class of modal logics than Kripke frames. Equivalently, establish whether ML(H) equals ML(FP) and whether ML(H) strictly contains ML(K), or identify which inclusions among ML(K) ⊆ ML(H) ⊆ ML(FP) are strict.

Background

Humberstone frames are introduced as a variant of possibility frames with stronger interplay conditions between refinement (\sqsubseteq) and accessibility (R), specifically up, R, and R, and with P equal to the full set of regular open sets. These frames lie between Kripke frames (the classical case) and the full possibility frames defined earlier in the paper.

The authors note that while every Humberstone frame is a full possibility frame and every Kripke frame is a Humberstone frame, it is not settled whether Humberstone frames are strictly more general than Kripke frames or whether they achieve the full generality of the possibility frames when it comes to characterizing normal modal logics. This motivates a precise comparison of ML(H), ML(K), and ML(FP).