On the Blok-Esakia theorem for universal classes

Abstract

The Blok-Esakia theorem states that there is an isomorphism from the lattice of intermediate logics onto the lattice of normal extensions of Grzegorczyk modal logic. The extension for multi-conclusion consequence relations was obtained by Emil Je\v{r}\'abek as an application of canonical rules. We show that Je\v{r}\'abek's result follows already from Blok's algebraic proof. We also prove that the properties of strong structural completeness and strong universal completeness are preserved and reflects by the aforementioned isomorphism. These properties coincide with structural completeness and universal completeness respectively for single-conclusion consequence relations and, in particular, for logics.