The universal modality, the center of a Heyting algebra, and the Blok-Esakia theorem

We introduce the bimodal logic S4.Grzu, which is the extension of Bennett’s bimodal logic S4u by Grzegorczyk’s axiom ( (p→ p)→ p)→ p and show that the lattice of normal extensions of the intuitionistic modal logic WS5 is isomorphic to the lattice of normal extensions of S4.Grzu, thus generalizing the Blok–Esakia theorem. We also introduce the intuitionistic modal logicWS5.C, which is the extension ofWS5 by the axiom ∀(p∨¬p)→ (p→ ∀p), and the bimodal logic S4.GrzuC, which is the extension of Shehtman’s bimodal logic S4uC by Grzegorczyk’s axiom, and show that the lattice of normal extensions ofWS5.C is isomorphic to the lattice of normal extensions of S4.GrzuC. © 2009 Elsevier B.V. All rights reserved.