Algebraic and Topological Methods in Non-Classical Logics II

Thanks also go to Professor Francesc Esteva, head of the Research Institute in Artificial Intelligence (IIIA) and to Professor`Angel Jorba, head of the Institute of Mathematics at the University of Barcelona (IMUB), for being so receptive to our needs and providing administrative support in different ways. The Oficina d'Activitats Institucionals of the University of Barcelona offered invaluable help coordinating several practical issues in the preparation of the meeting. The Organizing Committee is particularly indebted to Mrs. Pili Mateo for her efficiency. We also want to thank Mr. Pau Santanach for the excellent design he created for the meeting's logo and other graphic material, and Mrs. Montse Castilla of the IMUB for her professional help in handling the registration process and other administrative tasks. In this talk I will present a particular instance of fruitful use of algebraic and topological methods in non-classical logics by linking subframe logics with pointless topologies through nu-clei, which are unary operations on Heyting algebras satisfying certain identities. The advantages of this approach will be discussed in detail. What is the appropriate notion of " morphism " for general modal frames? This talk will give an answer by defining a notion of " modal map " between frames, generalizing the usual notion of bounded morphism/p-morphism, and reducing to it in the case of Kripke frames. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. The ultrafilter enlargement of a frame is the free compact Hausdorff frame generated by that frame relative to Fm. This free construction has an associated " monad " whose Eilenberg-Moore category is isomorphic to CHFm. An equivalence between the Kleisli category of the monad and UEFm can defined from a construction that assigns to each frame a unique image-closed frame with the same ultrafilter enlargement (an image-closed frame is one in which the set of alternatives of any point is topologically closed). These ideas are connected to a certain category shown by S. K. Thomason to be dual to the category of complete and atomic modal algebras and their homomorphisms. Thomason's category turns out to be the full subcategory of the above Kleisli category that is based on the Kripke frames. A modal logic is said …