  We extend unified correspondence theory to Kripke frames with impossible
worlds and their associated regular modal logics. These are logics the modal
connectives of which are not required to be normal: only the weaker properties
of additivity and multiplicativity are required. Conceptually, it has been
argued that their lacking necessitation makes regular modal logics better
suited than normal modal logics at the formalization of epistemic and deontic
settings. From a technical viewpoint, regularity proves to be very natural and
adequate for the treatment of algebraic canonicity J\'onsson-style. Indeed,
additivity and multiplicativity turn out to be key to extend J\'onsson's
original proof of canonicity to the full Sahlqvist class of certain regular
distributive modal logics naturally generalizing Distributive Modal Logic. Most
interestingly, additivity and multiplicativity are key to J\'onsson-style
canonicity also in the original (i.e. normal) DML. Our contributions include:
the definition of Sahlqvist inequalities for regular modal logics on a
distributive lattice propositional base; the proof of their canonicity
following J\'onsson's strategy; the adaptation of the algorithm ALBA to the
setting of regular modal logics on two non-classical (distributive lattice and
intuitionistic) bases; the proof that the adapted ALBA is guaranteed to succeed
on a syntactically defined class which properly includes the Sahlqvist one;
finally, the application of the previous results so as to obtain proofs,
alternative to Kripke's, of the strong completeness of Lemmon's epistemic
logics E2-E5 with respect to elementary classes of Kripke frames with
impossible worlds.
