Silvio Ghilardi (Università di Milano) will give a talk on
Investigations in profinite modal algebras and their dual Kripke frames
Abstract:
In [1] we showed that Prof(V) is monadic over Set and that its dual category is the category LfKFr(V) of locally finite Kripke frames validating V-equations (here V is any variety of modal algebras generated by its finite members). A couple of further research directions arise from these facts.
- Monadicity suggests the existence of an infinitary calculus for the propositional logic corresponding to V. In [2] we develop such a calculus and we relate some of its relevant metatheoretical properties (Craig interpolation and Beth definability) to exactness properties of the category LfKFr(V).
- LfKFr(V), being locally finitely presentable, is the category of models of an essentially algebraic theory T(V); this theory can be considered, in the locally finite case, a kind of `syntactic dual’ of the equational theory axiomatizing V. Giving a direct transparent description of such T(V) is a challenging task, revealing unexpected surprises, as it is evident form our solution presented in [3] for the case study of monadic Boolean algebras (= modal algebras for the modal system S5).
[1] M. De Berardinis, S. Ghilardi Profiniteness, Monadicity and Universal Models in Modal Logic, Annals of Pure and Applied Logic, 175(7), 103454, (2024);
[2] M. De Berardinis, S. Ghilardi A Proof Theory for Profinite Modal Algebras, arxiv preprint (2025);
[3] M. De Berardinis, S. Ghilardi An essentially algebraic glance to Kripke semantics: the S5 case, arxiv preprint (2025).