Algebraic Tools for Modal Logic

Monotone bounded distributive lattice expansions (DLMs) are boundeddistributive lattices augmented by nitary operations that are isotone orantitone in each coordinate. Such algebras encompass most algebras withbounded distributive lattice reducts that arise from logic, and general-ize bounded distributive lattices with operators. We de ne the canonicalextension for DLMs and show that this extension is functorial.A class of compatible DLMs is said to be canonical provided it isclosed under canonical extensions. The main theorems are two criteriafor canonicity: (1) If K is a canonical class of compatible DLMs closedunder ultraproducts, then the variety generated by K is also canonical; inparticular every nitely generated variety of DLMs is canonical. (2) Anyvariety of DLMs, for which the canonical extension of each basic operationof each DLM in the class is both continuous and dually continuous, iscanonical. Both of these preservation theorems encompass varieties notencompassed by previous results in the literature even in the setting ofBoolean algebras with additional operations.