Canonicity and Relativized Canonicity via Pseudo-Correspondence: an Application of ALBA

We generalize Venema’s result on the canonicity of the additivity of positive terms, from classical modal logic to a vast class of logics the algebraic semantics of which is given by varieties of normal distributive lattice expansions (normal DLEs), aka ‘distributive lattices with operators’. We provide two contrasting proofs for this result: the first is along the lines of Venema’s pseudocorrespondence argument but using the insights and tools of unified correspondence theory, and in particular the algorithm ALBA; the second closer to the style of Jónsson. Using insights gleaned from the second proof, we define a suitable enhancement of the algorithm ALBA, which we use prove the canonicity of certain syntactically defined classes of DLE-inequalities (called the metainductive inequalities), relative to the structures in which the formulas asserting the additivity of some given terms are valid.