Describable Nuclea, Negative Translations and Extension Stability

What do soundness (or completeness) of negative translations of logics with an intuitionistic base, extension stability of preservativity (or provability) logics, and subframe properties of modal logics have in common? As it turns out, in all those cases one deals with preservation of equalities in nucleic images of Heyting Algebra Expansions (HAEs). The nucleic perspective on subframe logics has been introduced by Silvio Ghilardi and Guram Bezhanishvili (APAL 2007) for the purely intuitionistic syntax without additional connectives. Since the 1970's, nuclei have been studied in the context of point-free topologies (lattice-complete Heyting algebras), sheaves and Grothendieck topologies on toposes, and finally arbitrary Heyting algebras (Macnab 1981). Other communities may know them as "lax modalities" or (the algebraic trace of) "strong monads". Nuclei are a rather trivial notion in the boolean setting, which partially explains why this approach has not been properly pursued in modal logic.
We marry the nuclear framework with that of "describable operations" introduced in Frank Wolter's PhD Thesis (1993). The latter was originally restricted to classical subframe logics, but with minimal it can be made to work in the Heyting setting and nuclea appear to provide the missing ingredient to utilize its full potential. From this perspective, we revisit our FSCD 2017 study of negative translations in modal logic (joint work with Miriam Polzer and Ulrich Rabenstein) and our present study of extension stability for preservativity logics based on the constructive strict implication (jointly with Albert Visser). Various characterization and completeness results can be obtained in a generic way.