Failure of the Blok-Esakia Theorem in the monadic setting

TL;DR

This paper demonstrates that the Blok-Esakia Theorem, which links superintuitionistic logics to Grzegorczyk's logic, fails to hold in the monadic fragment setting, showing the isomorphism does not extend.

Contribution

It proves that the Blok-Esakia isomorphism does not extend to the monadic fragments of the involved predicate logics, revealing limitations of the theorem.

Findings

The isomorphism fails in the monadic setting.

The lattice of extensions of monadic intuitionistic logic is not isomorphic to that of monadic Grzegorczyk logic.

The result highlights boundaries of the Blok-Esakia Theorem's applicability.

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.