Failure of the Blok-Esakia Theorem in the monadic setting

Abstract

The Blok-Esakia Theorem establishes that the lattice of superintuitionistic logics is isomorphic to the lattice of extensions of Grzegorczyk's logic. We prove that the Blok-Esakia isomorphism $\sigma$ does not extend to the fragments of the corresponding predicate logics of already one fixed variable. In other words, we prove that $\sigma$ is no longer an isomorphism from the lattice of extensions of the monadic intuitionistic logic to the lattice of extensions of the monadic Grzegorczyk logic.