Frame constructions, truth invariance and validity preservation in many-valued modal logic

In this paper we define and examine frame constructions for the family of many-valued modal logics introduced by M. Fitting in the ’90s. Every language of this family is built on an underlying space of truth values, a Heyting algebra H. We generalize Fitting’s original work by considering complete Heyting algebras as truth spaces and proceed to define a suitable notion of H-indexed families of generated subframes, disjoint unions and bounded morphisms. Then, we provide an algebraic generalization of the canonical extension of a frame and model, and prove a preservation result inspired from Fitting’s canonical model argument in [7]. The analog of a complex algebra and of a principal ultrafilter is defined and the embedding of a frame into its canonical extension is presented.